research article estimates for parameter littlewood-paley...
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Research ArticleEstimates for Parameter Littlewood-Paley 119892lowast
120581Functions on
Nonhomogeneous Metric Measure Spaces
Guanghui Lu and Shuangping Tao
College of Mathematics and Statistics Northwest Normal University Lanzhou 730070 China
Correspondence should be addressed to Shuangping Tao taospnwnueducn
Received 18 January 2016 Accepted 17 March 2016
Academic Editor Yoshihiro Sawano
Copyright copy 2016 G Lu and S Tao This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Let (X 119889 120583) be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measureconditions In this paper the authors prove that under the assumption that the kernel of Mlowast
120581satisfies a certain Hormander-type
condition Mlowast120588
120581is bounded from Lebesgue spaces 119871119901
(120583) to Lebesgue spaces 119871119901
(120583) for 119901 ge 2 and is bounded from 1198711
(120583) into1198711infin
(120583) As a corollary Mlowast120588
120581is bounded on 119871
119901
(120583) for 1 lt 119901 lt 2 In addition the authors also obtain that Mlowast120588
120581is bounded from
the atomic Hardy space1198671
(120583) into the Lebesgue space 1198711
(120583)
1 Introduction
In 1958 Stein in [1] firstly introduced the Littlewood-Paleyoperators of the higher-dimensional case meanwhile theauthor also obtained the boundedness of the Marcinkiewiczintegrals and area integrals In 1970 Fefferman in [2] provedthat the Littlewood-Paley 119892lowast
120581function is weak type (119901 119901) for
119901 isin (1 2) and 120581 = 2119901 With further research about Little-wood-Paley operators some authors turn their attentions tostudy the parameter Littlewood-Paley operators For exam-ple in 1999 Sakamoto and Yabuta in [3] considered theparameter 119892lowast
120581function Since then many papers focus on the
behaviours of the operators among them we refer readers tosee [4ndash6]
In the past ten years or so most authors mainly study theclassical theory of harmonic analysis on R119899 under nondou-bling measures which only satisfy the polynomial growthcondition see [7ndash12] Exactly we assume that 120583 which is apositive Radon measure on R119899 satisfies the following growthconditions namely for all 119909 isin R119899 and 119903 isin (0infin) there existconstant 119862 and 0 lt 119889 le 119899 such that
120583 (119861 (119909 119903)) le 119862119903119889
(1)
where 119861(119909 119903) fl 119910 isin R119899
|119909 minus 119910| lt 119903 The analysisassociated with nondoubling measures 120583 as in (1) hasimportant applications in solving long-standing open Pain-leversquos problem and Vitushkinrsquos conjecture (see [13 14])Besides Coifman andWeiss have showed that the measure 120583is a key assumption in harmonic analysis on homogeneous-type spaces (see [15 16])
HoweverHytonen in [17] pointed that themeasure120583 as in(1) may not contain the doubling measure as special cases Tosolve the problem in 2010 Hytonen in [17] introduced a newclass of metric measure spaces satisfying the so-called upperdoubling conditions and the geometrically doubling (respsee Definitions 1 and 2 below) which are now claimed non-homogeneousmetricmeasure spacesTherefore if we replacethe underlying spaceswith nonhomogeneousmetricmeasurespaces many known-consequences have been proved stilltrue for example see [18ndash22]
In this paper we always assume that (X 119889 120583) is a non-homogeneous metric measure space In this setting we willestablish the boundedness of the parameter Littlewood-Paley119892lowast
120581functions on (X 119889 120583)In order to state our main results we firstly recall some
necessary notions and notation Hytonen in [17] gave out thedefinition of upper doubling metric spaces as follows
Hindawi Publishing CorporationJournal of Function SpacesVolume 2016 Article ID 9091478 12 pageshttpdxdoiorg10115520169091478
2 Journal of Function Spaces
Definition 1 (see [17]) A metric measure space (X 119889 120583) issaid to be upper doubling if 120583 is Borel measure on X andthere exist a dominating function 120582 X times (0infin) rarr (0infin)
and a positive constant 119862120582such that for each 119909 isin X 119903 rarr
120582(119909 119903) is nondecreasing and for all 119909 isin X and 119903 isin (0infin)
120583 (119861 (119909 119903)) le 120582 (119909 119903) le 119862120582120582 (119909
119903
2
) (2)
Htyonen et al in [18] proved that there exists anotherdominating function
120582 such that 120582 le 120582 119862120582le 119862
120582and
120582 (119909 119910) le 119862
120582
120582 (119910 119903) (3)
where 119909 119910 isin X and 119889(119909 119910) le 119903 Based on this from now onlet the dominating function in (2) also satisfy (3)
Now we recall the notion of geometrically doubling con-ditions given in [17]
Definition 2 (see [17]) A metric space (X 119889) is said to begeometrically doubling if there exists some 119873
0isin N such
that for any ball 119861(119909 119903) sub X there exists a finite ball cover-ing 119861(119909
119894 1199032)
119894of 119861(119909 119903) such that the cardinality of this
covering is at most1198730
Remark 3 (see [17]) Let (X 119889) be a metric space Hytonen in[17] showed that the following statements aremutually equiv-alent
(1) (X 119889) is geometrically doubling
(2) For any 120598 isin (0 1) and ball 119861(119909 119903) sub X there exists afinite ball covering 119861(119909
119894 120598119903)
119894of 119861(119909 119903) such that the
cardinality of this covering is at most1198730120598minus119899 Here and
in what follows1198730is as Definition 2 and 119899 = log
2119873
0
(3) For every 120598 isin (0 1) any ball 119861(119909 119903) sub X can containat most119873
0120598minus119899 centers of disjoint balls 119861(119909
119894 120598119903)
119894
(4) There exists 119872 isin N such that any ball 119861(119909 119903) sub X
can contain at most 119872 centers 119909119894119894of disjoint balls
119861(119909119894 1199034)
119872
119894=1
Hytonen in [17] introduced the following coefficients119870119861119878
analogous to Tolsarsquos number 119870119876119877
in [7]Given any two balls 119861 sub 119878 set
119870119861119878
fl 1 + int
2119878119861
1
120582 (119888119861 119889 (119909 119888
119861))
d120583 (119909) (4)
where 119888119861represents the center of the ball 119861
Remark 4 Bui and Duong in [21] firstly introduced the fol-lowing discrete version
119870119861119878
of 119870119861119878
as in (4) on (X 119889 120583)
which is very similar to the number119870119876119877
introduced in [7] byTolsa For any two balls 119861 sub 119878 119870
119861119878is defined by
119870
119861119878= 1 +
119873119861119878
sum
119894=1
120583 (6119894
119861)
120582 (119888119861 6
119894119903119861)
(5)
where the radii of the balls 119861 and 119878 are denoted by 119903119861and
119903119878 respectively and 119873
119861119878is the smallest integer satisfying
6119873119861119878
119903119861ge 119903
119904 It is easy to obtain
119870119861119878
le 119862119870119861119878 Bui and Duong
in [21] also pointed out that it is incorrect that119870119861119878
sim119870
119861119878
Now we recall the following notion of (120572 120573)-doublingproperty (see [17])
Definition 5 (see [17]) Let 120572 120573 isin (1infin) A ball 119861 sub X isclaimed to be (120572 120573)-doubling if 120583(120572119861) le 120573120583(119861)
It was stated in [17] that there exist many balls whichhave the above (120572 120573)-doubling property In the latter part ofthe paper if 120572 and 120573
120572are not specified (120572 120573
120572)-doubling ball
always stands for (6 1205736)-doubling ball with a fixed number
1205736gt max1198623 log
26
120582 6
119899
where 119899 fl log2119873
0is considered as
a geometric dimension of the space Moreover the smallest(6 120573
6)-doubling ball of the form 6
119895
119861 with 119895 isin N is denotedby
119861
6 and sometimes 1198616 can be simply denoted by 119861
Now we give the definition of the parameter Littlewood-Paley 119892lowast
120581functions on (X 119889 120583)
Definition 6 (see [22]) Let 119870(119909 119910) be a locally integrablefunction on (X times X) (119909 119910) 119909 = 119910 Assume that thereexists a positive constant 119862 such that for all 119909 119910 isin X with119909 = 119910
1003816100381610038161003816119870 (119909 119910)
1003816100381610038161003816le 119862
119889 (119909 119910)
120582 (119909 119889 (119909 119910))
(6)
and for all 119909 119910 1199101015840
isin X
int
119889(119909119910)ge2119889(1199101199101015840)
[
10038161003816100381610038161003816119870 (119909 119910) minus 119870 (119909 119910
1015840
)
10038161003816100381610038161003816
+
10038161003816100381610038161003816119870 (119910 119909) minus 119870 (119910
1015840
119909)
10038161003816100381610038161003816]
1
119889 (119909 119910)
d120583 (119909) le 119862
(7)
The parameter Marcinkiewicz integral M120588 associatedwith the above 119870(119909 119910) which satisfies (6) and (7) is definedby
M120588
(119891) (119909) = (int
infin
0
1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
sdot int
119889(119909119910)le119905
119870(119909 119910)
[119889 (119909 119910)]
1minus120588119891 (119910) d120583 (119910)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d119905119905
)
12
119909 isin X
(8)
Journal of Function Spaces 3
where 120588 isin (0infin)The parameter 119892lowast
120581functionMlowast120588
120581is defined
by
Mlowast120588
120581(119891) (119909) = ∬
Xtimes(0infin)
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
sdot int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910)
120582 (119910 119905)
d119905119905
12
(9)
where 119909 isin X X times (0infin) fl (119910 119905) 119910 isin X 119905 gt 0 120588 gt 0
and 120581 isin (1infin)
Remark 7 (1) When 120588 = 1 the operator M120588 as in (8) is justthe Marcinkiewicz integral on (X 119889 120583) (see [22])
(2) If we take (X 119889 120583) = (R119899
|sdot| 120583) and 120582(119910 119905) fl 119905119899 then
the parameter 119892lowast
120581functionMlowast120588
120581as in (9) is just a parameter
Littlewood-Paley operatorwith nondoublingmeasures in [8]
The following definition of the atomic Hardy space wasintroduced by Htyonen et al (see [18])
Definition 8 (see [18]) Let 120577 isin (1infin) and 119901 isin (1infin] Afunction 119887 isin 119871
1
loc(120583) is called a (119901 1)120592-atomic block if
(a) there exists a ball 119861 such that supp 119887 sub 119861
(b) intX119887(119909)d120583(119909) = 0
(c) for any 119894 isin 1 2 there exist a function 119886119894supported
on ball 119861119894sub 119861 and a number 120592
119894isin C such that
119887 = 12059211198861+ 120592
21198862
1003817100381710038171003817119886119894
1003817100381710038171003817119871119901(120583)
le [120583 (120577119861119894)]
1119901minus1
119870minus1
119861119894 119861
(10)
Moreover let |119887|119867
1119901
atb (120583)fl |120592
1| + |120592
2|
We say a function 119891 isin 1198711
(120583) belongs to the atomicHardy space 119867
1119901
atb (120583) if there are atomic blocks 119887119894infin
119894=1such
that 119891 = suminfin
119894=1119887119894with sum
infin
119894=1|119887
119894|119867
1119901
atb (120583)lt infin The 1198671119901
atb (120583) normof 119891 is denoted by 119891
119867
1119901
atb (120583)= infsuminfin
119894=1|119887
119894|119867
1119901
atb (120583) where the
infimum is taken over all the possible decompositions of 119891 asabove
It was proved by Htyonen et al in [18] that the definitionof 1198671119901
atb (120583) is not related to the choice of 120577 and the spaces119867
1119901
atb (120583) and 1198671infin
atb (120583) have the same norms for 119901 isin (1infin]Thus for convenience we always denote1198671119901
atb (120583) by1198671
(120583)Nowwe give the Hormander-type condition on (X 119889 120583)
that is there exists a positive 119862 such that
sup119903gt0
119889(1199101199101015840)le119903
infin
sum
119894=1
119894 int
6119894119903lt119889(119909119910)le6
119894+1119903
[
10038161003816100381610038161003816119870 (119909 119910) minus 119870 (119909 119910
1015840
)
10038161003816100381610038161003816+
10038161003816100381610038161003816119870 (119910 119909) minus 119870 (119910
1015840
119909)
10038161003816100381610038161003816]
d120583 (119909)
119889 (119909 119910)
le 119862 (11)
Notice this condition is slightly stronger than (7)Now let us state the main theorems which generalize and
improve the corresponding results in [8]
Theorem 9 Let119870(119909 119910) satisfy (6) and (7) and letMlowast120588
120581be as
in (9) with 120588 isin (0infin) and 120581 isin (1infin) ThenMlowast120588
120581is bounded
on 119871119901
(120583) for any 119901 isin [2infin)
Theorem 10 Let 119870(119909 119910) satisfy (6) and (11) and letMlowast120588
120581be
as in (9) with 120588 isin (12infin) and 120581 isin (1infin) Then Mlowast120588
120581is
bounded from 1198711
(120583) into weak 1198711
(120583) namely there exists apositive constant 119862 such that for any 120591 gt 0 and 119891 isin 119871
1
(120583)
120583 (119909 isin X Mlowast120588
120581(119891) (119909) gt 120591) le 119862
100381710038171003817100381711989110038171003817100381710038171198711(120583)
120591
(12)
Theorem 11 Let 119870(119909 119910) satisfy (6) and (11) and let Mlowast120588
120581be
as in (9) with 120588 gt 12 and 120581 gt 1 Suppose thatMlowast120588
120581is bounded
on 1198712
(120583) ThenMlowast120588
120581is bounded from119867
1
(120583) into 1198711
(120583)
Applying the Marcinkiewicz interpolation theorem andTheorems 9 and 10 it is easy to get the following result
Corollary 12 Under the assumption of Theorem 10 Mlowast120588
120581is
bounded on 119871119901
(120583) for 119901 isin (1 2)
The organization of this paper is as follows In Section 2wewill give somepreliminary lemmasTheproofs of themaintheorems will be given in Section 3 Throughout this paper119862 stands for a positive constant which is independent of themain parameters but it may be different from line to line Forany 119864 sub X we use 120594
119864to denote its characteristic function
2 Preliminary Lemmas
In this section we make some preliminary lemmas which areused in the proof of the main results Firstly we recall someproperties of119870
119861119878as in (4) (see [17])
Lemma 13 (see [17]) (1) For all balls 119861 sub 119877 sub 119878 it holds truethat 119870
119861119877le 119870
119861119878
(2) For any 120585 isin [1infin) there exists a positive constant 119862120585
such that for all balls 119861 sub 119878 with 119903119878le 120585119903
119861 119870
119861119878le 119862
120585
(3) For any 984858 isin (1infin) there exists a positive constant 119862984858
depending on 984858 such that for all balls 119861119870119861
119861
984858 le 119862984858
(4) There exists a positive constant 119888 such that for all balls119861 sub 119877 sub 119878 119870
119861119878le 119870
119861119877+ 119888119870
119877119878 In particular if 119861 and 119877 are
concentric then 119888 = 1(5) There exists a positive constant such that for all balls
119861 sub 119877 sub 119878 119870119861119877
le 119870119861119878 moreover if 119861 and 119877 are concentric
then 119870119877119878
le 119870119861119878
4 Journal of Function Spaces
To state the following lemmas let us give a known-result(see [19]) For 120578 isin (0infin) the maximal operator is defined bysetting that for all 119891 isin 119871
1
loc(120583) and 119909 isin X
119872(120578)119891 (119909) fl sup
119876ni119909119876doubling
1
120583 (120578119876)
int
119876
1003816100381610038161003816119891 (119910)
1003816100381610038161003816d120583 (119910) (13)
is bounded on 119871119901
(120583) provided that 119901 isin (1infin) and alsobounded from 119871
1
(120583) into 1198711infin
(120583)The following lemma is slightly changed from [8]
Lemma 14 Let 119870(119909 119910) satisfy (6) and (7) and 120578 isin (0infin)Assume that M120588 is as in (8) and Mlowast120588
120581is as in (9) with
120588 isin (0infin) and 120581 isin (1infin) Then for any nonnegative function120601 there exists a positive constant 119862 such that for all 119891 isin 119871
119901
(120583)
with 119901 isin (1infin)
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
le 119862int
X
[M120588
(119891) (119909)]
2
119872120578(120601) (119909) d120583 (119909)
(14)
Proof By the definition ofMlowast120588
120581(119891) we have
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
= int
X
∬
Xtimes(0infin)
(
119905
119905 + 119889 (119909 119910)
)
120573 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588119891 (119910) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910)
120582 (119910 119905)
d119905119905
120601 (119909) d120583 (119909)
le int
X
int
infin
0
1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119910) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d119905119905
sup119905gt0
[int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)] d120583 (119910)
= int
X
[M120588
(119891) (119910)]
2 sup119905gt0
[int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)] d120583 (119910)
(15)
Thus to prove Lemma 14 we only need to estimate that
sup119905gt0
int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
le 119862119872120578(120601) (119910)
(16)
For any 119910 isin X and 119905 gt 0 write
int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
= int
119861(119910119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
+ int
X119861(119910119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
š 1198631+ 119863
2
(17)
For1198631 it is not difficult to obtain that
1198631le int
119861(119910119905)
120601 (119909)
120582 (119910 119905)
d120583 (119909)
=
120583 (120578119861 (119910 119905))
120582 (119910 119905)
1
120583 (120578119861 (119910 119905))
int
119861(119910119905)
120601 (119909) d120583 (119909)
le 119862119872120578(120601) (119910)
(18)
Now we turn to estimate1198632 by (2) and (13) we have
1198632le
infin
sum
119896=1
int
119861(1199106119896119905)119861(1199106
119896minus1119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
sdot
120601 (119909)
120582 (119910 119905)
d120583 (119909) le 119862
infin
sum
119896=1
6minus(119896minus1)120573
sdot int
119861(1199106119896119905)
120601 (119909)
120582 (119910 119905)
d120583 (119909) le 119862
infin
sum
119896=1
6minus(119896minus1)120573
sdot
120583 (119861 (119910 6119896
119905))
120582 (119910 119905)
1
120583 (119861 (119910 6119896119905))
int
119861(1199106119896119905)
120601 (119909) d120583 (119909)
le 119862
infin
sum
119896=1
6minus(119896minus1)120573
120583 (119861 (119910 6119896
119905))
120582 (119910 119905)
119872120578(120601) (119910) le 119862
sdot
120582 (119910 6119896
119905)
120582 (119910 119905)
119872120578(120601) (119910)
infin
sum
119896=1
6minus(119896minus1)120573
120582 (119910 6119896
119905)
120582 (119910 119905)
le 119862
sdot
120582 (119910 6119896
119905)
120582 (119910 119905)
119872120578(120601) (119910)
infin
sum
119896=1
6minus(119896minus1)120573
le 119862119872120578(120601) (119910)
(19)
Combining the estimates for 1198631and 119863
2 we obtain (16) and
hence complete the proof of Lemma 14
Finally we recall the Calderon-Zygmund decompositiontheorem (see [21]) Suppose that 120574
0is a fixed positive constant
Journal of Function Spaces 5
satisfying that 1205740gt max1198623 log
26
120582 6
3119899
where 119862120582is as in (2)
and 119899 as in Remark 3
Lemma 15 (see [21]) Let 119901 isin [1infin) 119891 isin 119871119901
(120583) and 119905 isin
(0infin) (119905 gt 1205740119891
119871119901(120583)
120583(X) when 120583(X) lt infin) Then
(1) there exists a family of finite overlapping balls 6119861119894119894
such that 119861119894119894is pairwise disjoint
1
120583 (62119861119894)
int
119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119889120583 (119909) gt
119905119901
1205740
forall119894 (20)
1
120583 (62120591119861
119894)
int
120591119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119889120583 (119909) le
119905119901
1205740
forall119894 forall120591 isin (2infin)
(21)
1003816100381610038161003816119891 (119909)
1003816100381610038161003816le 119905
119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X (⋃
119894
6119861119894)
(22)
(2) for each 119894 let 119878119894be a (3 times 6
2
119862
log2(3times62)+1
120582)-doubling ball
of the family (3times62
)119896
119861119894119896isinN and 120596119894
= 1205946119861119894
(sum1198961205946119861119896
)Then there exists a family 120593
119894119894of functions that for
each 119894 supp(120593119894) sub 119878
119894 120593
119894has a constant sign on 119878
119894and
int
X
120593119894(119909) 119889120583 (119909) = int
6119861119894
119891 (119909) 120596119894(119909) 119889120583 (119909)
sum
119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816le 120574119905 119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X
(23)
where 120574 is some positive constant depending only on (X 120583)and there exists a positive constant 119862 independent of 119891 119905 and119894 such that if 119901 = 1 then
1003817100381710038171003817120593119894
1003817100381710038171003817119871infin(120583)
120583 (119878119894) le 119862int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816119889120583 (119909) (24)
and if 119901 isin (1infin)
(int
119878119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816
119901
119889120583 (119909))
1119901
[120583 (119878119894)]
11199011015840
le
119862
119905119901minus1
int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816
119901
119889120583 (119909)
(25)
3 Proofs of Theorems
Proof ofTheorem 9 For the case of 119901 = 2 assume 120601(119909) = 1 inLemma 14 then it is easy to get that
int
X
[Mlowast120588
120581(119891) (119909)]
2 d120583 (119909)
le 119862int
X
[M120588
(119891) (119909)]
2 d120583 (119909)
(26)
which along with 1198712
(120583)-boundedness of M120588 easily yieldsthat Theorem 9 holds
For the case of 119901 gt 2 let 119902 be the index conjugate to1199012 By applying Holder inequality and Lemma 14 we canconclude
1003817100381710038171003817M
lowast120588
120581(119891)
1003817100381710038171003817
2
119871119901(120583)
= sup120601ge0
120601119871119902(120583)
le1
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
le 119862 sup120601ge0
120601119871119902(120583)
le1
int
X
[M120588
(119891) (119909)]
2
119872120578120601 (119909) d120583 (119909)
le 1198621003817100381710038171003817M
120588
(119891)1003817100381710038171003817
2
119871119901(120583)
sup120601ge0
120601119871119902(120583)
le1
10038171003817100381710038171003817119872
120578(120601)
10038171003817100381710038171003817119871119902(120583)
le 11986210038171003817100381710038171198911003817100381710038171003817
2
119871119901(120583)
sup120601ge0
120601119871119902(120583)
le1
10038171003817100381710038171206011003817100381710038171003817119871119902(120583)
le 11986210038171003817100381710038171198911003817100381710038171003817
2
119871119901(120583)
(27)
which is desired Thus we complete the proof of Theorem 9
Proof of Theorem 10 Without loss of generality we mayassume that 119891
1198711(120583)
= 1 It is easy to see that the conclusionofTheorem 10 naturally holds if 120591 le 120573
6(119891
1198711(120583)
120583(X)) when120583(X) lt infin Thus we only need to discuss the case that120591 gt 120573
6(119891
1198711(120583)
120583(X)) Applying Lemma 15 to119891 at the level 120591and letting120596
119894120593
119894119861
119894 and 119878
119894be the same as in Lemma 15 we see
that119891(119909) = 119887(119909)+ℎ(119909) where 119887(119909) fl 119891120594X⋃1198946119861119894
(119909)+sum119894120593119894(119909)
and ℎ(119909) fl sum119894[120596
119894(119909)119891(119909) minus 120593
119894(119909)] š sum
119894ℎ119894(119909) It is easy
to obtain that 119887119871infin(120583)
le 119862120591 and 1198871198711(120583)
le 119862 By 1198712
(120583)-boundedness ofMlowast120588
120581 we have
120583 (119909 isin X Mlowast120588
120581(119887) (119909) gt 120591) le
1003817100381710038171003817Mlowast120588
120581(119887)
1003817100381710038171003817
2
1198712(120583)
1205912
le 119862
1198872
1198712(120583)
1205912
le 119862120591minus1
(28)
On the other hand by (20) with 119901 = 1 and the fact that thesequence of balls 119861
119894119894 is pairwise disjoint we see that
120583(⋃
119894
62
119861119894) le 119862120591
minus1
int
X
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) le 119862120591
minus1
(29)
and thus the proof of theTheorem 10 can be reduced to provethat
120583(119909 isin X ⋃
119894
62
119861119894 M
lowast120588
120581(ℎ) (119909) gt 120591) le 119862120591
minus1
(30)
For each fixed 119894 denote the center of 119861119894by 119909
119894 and let 119873
1
be the positive integer satisfying 119878119894= (3 times 6
2
)1198731119861119894 We have
120583(119909 isin X ⋃
119894
62
119861119894 M
lowast120588
120581(ℎ) (119909) gt 120591)
le 120591minus1
sum
119894
int
X⋃11989462119861119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
6 Journal of Function Spaces
le 120591minus1
sum
119894
int
X6119878119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
+ 120591minus1
sum
119894
int
611987811989462119861119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
š 120591minus1
sum
119894
(1198641+ 119864
2)
(31)
Firstly let us estimate 1198642and write it as
1198642le int
611987811989462119861119894
Mlowast120588
120581(119891120596
119894) (119909) d120583 (119909)
+ int
611987811989462119861119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909) š 119864
21+ 119864
22
(32)
where ℎ119894fl 120596
119894119891 minus 120593
119894 By Holder inequality (24) and 119871
2
(120583)-boundedness ofMlowast120588
120581 we have
11986422
le int
6119878119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909)
le (int
6119878119894
1003816100381610038161003816M
lowast120588
120581(120593
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862(int
6119878119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816d120583 (119909)
(33)
For 11986421 by Minkowski inequality and (6) write
11986421
= int
611987811989462119861119894
[
[
∬
Xtimes(0infin)
1003816100381610038161003816100381610038161003816100381610038161003816
(
119905
119905 + 119889 (119909 119910)
)
1205812
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119911) 120596119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
le 119862int
611987811989462119861119894
int
X
1003816100381610038161003816119891 (119911) 120596
119894(119911)
1003816100381610038161003816[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905
2119889(119910119911)gt119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)lt119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
š 1198651+ 119865
2+ 119865
3
(34)
To this end let 119861119894be as in Lemma 15 with 119888
119861119894and 119903
119861119894being
respectively its center and radius For any 119909 isin 6119878119894 6
2
119861119894and
119911 isin 6119861119894 by (2) and (3) we have
1198651le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
int
infin
119889(119910119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) dt120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119911))
(int
infin
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119910 119911))]
3
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
Journal of Function Spaces 7
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
d120583 (119910)
[120582 (119910 (12) 119889 (119909 119911))]
2
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 (12) 119889 (119909 119911))]2int
2119889(119910119911)gt119889(119909119911)
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))119861(1199112119896minus2
119889(119909119911))
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
1
120582 (119911 119889 (119909 119911))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))
d120583 (119910)
120582 (119910 2119896minus2
119889 (119909 119911))
]
12
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(35)
where we use the fact that
int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) le 119862119870119861119894 119878119894
(36)
Next we estimate 1198652 For any 119909 isin 6119878
119894 6
2
119861119894 119910 isin X
and 119911 isin 6119861119894satisfying 119889(119910 119909) lt 119905 2119889(119910 119911) le 119889(119909 119911) and
(12)119889(119909 119911) lt 119905 we have
1198652le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
int
infin
(12)119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
d120583 (119910)
120583 (119861 (119910 119889 (119909 119911)))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(37)
Finally for any 119909 isin 6119878119894 6
2
119861119894 119910 isin X and 119911 isin 6119861
119894sat-
isfying 2119889(119910 119911) le 119889(119909 119911) 2119889(119910 119911) ge 119889(119909 119911) and 119889(119909 119910) lt
(32)119889(119909 119911) by applying (2) we have
1198653le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
1
120582 (119910 119889 (119909 119911))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 119889 (119909 119911))]2
120583 (119861 (119911 (12) 119889 (119909 119911)))
120582 (119911 119889 (119909 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(38)
8 Journal of Function Spaces
Combining the estimates for 1198651 119865
2 and 119865
3 we obtain that
11986421
le 119862int6119861119894
|119891(119911)|d120583(119911) where together with the fact that119864
22le 119862int
6119861119894
|119891(119911)|d120583(119911) we have
1198642le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (39)
Now we turn to estimate for 1198641 Let 119876
119894= 119861(119888
119861119894 119903
119878119894) and
write
1198641le int
X6119878119894
[∬
119889(119909119910)lt119905
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isin119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isinX119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 11986411
+ 11986412
+ 11986413
(40)
For each fixed 119894 decompose 11986411as
11986411
le int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isin2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isinX2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 1198681+ 119868
2
(41)
For any 119909 isin X 6119878119894 119910 isin 2119878
119894with 119889(119910 119909) lt 119905 and
119911 isin 119878119894 119889(119909 119888
119861119894) minus 2119903
119878119894le 119889(119909 119910) lt 119905 and 119889(119910 119911) lt 3119903
119878119894
togetherwithMinkowski inequality and (6) we can conclude
1198681le 119862int
X6119878119894
int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
∬119889(119909119910)lt119905119889(119910119911)le119905
119910isin2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
(int
infin
119889(119909119888119861119894)minus2119903119878119894
d119905120582 (119910 119905) 119905
1+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
1
[119889 (119909 119888119861119894) minus 2119903
119878119894]
2120588
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(42)
Journal of Function Spaces 9
For 1198682 write
1198682le int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905le119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
š 11986821
+ 11986822
(43)
For 11986821 by Minkowski inequality and (6) we deduce
11986821
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
119889(119910119888119861119894)+119903119878119894
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[int
2119896+1
611987811989421198966119878119894
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(44)
Nowwe estimate 11986822 ApplyingMinkowski inequality and the
vanishing moment we have
11986822
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870(119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588+
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
+ 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909) š 1198691+ 119869
2
(45)
With a way similar to that used in the proof of 1198681 we have
1198691le 119862ℎ
1198941198711(120583) Thus we only need to estimate 119869
2 by Mink-
owski inequality and (11) it follows that
1198692le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2minus2120588
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
infin
119889(119910119888119861119894)+119903119878119894
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
10 Journal of Function Spaces
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[
[
int
2119896119903119878119894
lt119889(1199101198881198611)le2119896+1
119903119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2d120583 (119910)
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(46)
Combining the estimates for 1198691 119869
2 119868
21 and 119868
1 we obtain that
11986411
le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(47)
Next we estimate 11986412 For any 119910 isin 119861
119894 119909 isin X 6119878
119894 and
119911 isin 119878119894 we have 119889(119909 119910) ge (12)119889(119909 119888
119861119894) 119889(119910 119911) le 2119903
119878119894 and
119889(119909 119910) sim 119889(119909 119888119861119894) and together with this fact Minkowski
inequality and (6) we get
11986412
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
[
∬119889(119909119910)ge119905
119889(119910119911)le119905
119910isin119876119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(48)
It remains to estimate 11986413 Applying Minkowski inequality
and (6) we have
11986413
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
+ 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)gt2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
š 1198801+ 119880
2
(49)
Journal of Function Spaces 11
Now we estimate 1198801 For any 119910 isin X 119876
119894 119911 isin 119878
119894 and 119889(119910
119911) le 119905 le 119889(119909 119911) it is easy to see 119889(119910 119911) sim 119889(119910 119888119861119894) So we have
1198801le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
d120583 (119910)
120582 (119910 119889 (119910 119888119861119894))
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(50)
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Function Spaces
Definition 1 (see [17]) A metric measure space (X 119889 120583) issaid to be upper doubling if 120583 is Borel measure on X andthere exist a dominating function 120582 X times (0infin) rarr (0infin)
and a positive constant 119862120582such that for each 119909 isin X 119903 rarr
120582(119909 119903) is nondecreasing and for all 119909 isin X and 119903 isin (0infin)
120583 (119861 (119909 119903)) le 120582 (119909 119903) le 119862120582120582 (119909
119903
2
) (2)
Htyonen et al in [18] proved that there exists anotherdominating function
120582 such that 120582 le 120582 119862120582le 119862
120582and
120582 (119909 119910) le 119862
120582
120582 (119910 119903) (3)
where 119909 119910 isin X and 119889(119909 119910) le 119903 Based on this from now onlet the dominating function in (2) also satisfy (3)
Now we recall the notion of geometrically doubling con-ditions given in [17]
Definition 2 (see [17]) A metric space (X 119889) is said to begeometrically doubling if there exists some 119873
0isin N such
that for any ball 119861(119909 119903) sub X there exists a finite ball cover-ing 119861(119909
119894 1199032)
119894of 119861(119909 119903) such that the cardinality of this
covering is at most1198730
Remark 3 (see [17]) Let (X 119889) be a metric space Hytonen in[17] showed that the following statements aremutually equiv-alent
(1) (X 119889) is geometrically doubling
(2) For any 120598 isin (0 1) and ball 119861(119909 119903) sub X there exists afinite ball covering 119861(119909
119894 120598119903)
119894of 119861(119909 119903) such that the
cardinality of this covering is at most1198730120598minus119899 Here and
in what follows1198730is as Definition 2 and 119899 = log
2119873
0
(3) For every 120598 isin (0 1) any ball 119861(119909 119903) sub X can containat most119873
0120598minus119899 centers of disjoint balls 119861(119909
119894 120598119903)
119894
(4) There exists 119872 isin N such that any ball 119861(119909 119903) sub X
can contain at most 119872 centers 119909119894119894of disjoint balls
119861(119909119894 1199034)
119872
119894=1
Hytonen in [17] introduced the following coefficients119870119861119878
analogous to Tolsarsquos number 119870119876119877
in [7]Given any two balls 119861 sub 119878 set
119870119861119878
fl 1 + int
2119878119861
1
120582 (119888119861 119889 (119909 119888
119861))
d120583 (119909) (4)
where 119888119861represents the center of the ball 119861
Remark 4 Bui and Duong in [21] firstly introduced the fol-lowing discrete version
119870119861119878
of 119870119861119878
as in (4) on (X 119889 120583)
which is very similar to the number119870119876119877
introduced in [7] byTolsa For any two balls 119861 sub 119878 119870
119861119878is defined by
119870
119861119878= 1 +
119873119861119878
sum
119894=1
120583 (6119894
119861)
120582 (119888119861 6
119894119903119861)
(5)
where the radii of the balls 119861 and 119878 are denoted by 119903119861and
119903119878 respectively and 119873
119861119878is the smallest integer satisfying
6119873119861119878
119903119861ge 119903
119904 It is easy to obtain
119870119861119878
le 119862119870119861119878 Bui and Duong
in [21] also pointed out that it is incorrect that119870119861119878
sim119870
119861119878
Now we recall the following notion of (120572 120573)-doublingproperty (see [17])
Definition 5 (see [17]) Let 120572 120573 isin (1infin) A ball 119861 sub X isclaimed to be (120572 120573)-doubling if 120583(120572119861) le 120573120583(119861)
It was stated in [17] that there exist many balls whichhave the above (120572 120573)-doubling property In the latter part ofthe paper if 120572 and 120573
120572are not specified (120572 120573
120572)-doubling ball
always stands for (6 1205736)-doubling ball with a fixed number
1205736gt max1198623 log
26
120582 6
119899
where 119899 fl log2119873
0is considered as
a geometric dimension of the space Moreover the smallest(6 120573
6)-doubling ball of the form 6
119895
119861 with 119895 isin N is denotedby
119861
6 and sometimes 1198616 can be simply denoted by 119861
Now we give the definition of the parameter Littlewood-Paley 119892lowast
120581functions on (X 119889 120583)
Definition 6 (see [22]) Let 119870(119909 119910) be a locally integrablefunction on (X times X) (119909 119910) 119909 = 119910 Assume that thereexists a positive constant 119862 such that for all 119909 119910 isin X with119909 = 119910
1003816100381610038161003816119870 (119909 119910)
1003816100381610038161003816le 119862
119889 (119909 119910)
120582 (119909 119889 (119909 119910))
(6)
and for all 119909 119910 1199101015840
isin X
int
119889(119909119910)ge2119889(1199101199101015840)
[
10038161003816100381610038161003816119870 (119909 119910) minus 119870 (119909 119910
1015840
)
10038161003816100381610038161003816
+
10038161003816100381610038161003816119870 (119910 119909) minus 119870 (119910
1015840
119909)
10038161003816100381610038161003816]
1
119889 (119909 119910)
d120583 (119909) le 119862
(7)
The parameter Marcinkiewicz integral M120588 associatedwith the above 119870(119909 119910) which satisfies (6) and (7) is definedby
M120588
(119891) (119909) = (int
infin
0
1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
sdot int
119889(119909119910)le119905
119870(119909 119910)
[119889 (119909 119910)]
1minus120588119891 (119910) d120583 (119910)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d119905119905
)
12
119909 isin X
(8)
Journal of Function Spaces 3
where 120588 isin (0infin)The parameter 119892lowast
120581functionMlowast120588
120581is defined
by
Mlowast120588
120581(119891) (119909) = ∬
Xtimes(0infin)
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
sdot int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910)
120582 (119910 119905)
d119905119905
12
(9)
where 119909 isin X X times (0infin) fl (119910 119905) 119910 isin X 119905 gt 0 120588 gt 0
and 120581 isin (1infin)
Remark 7 (1) When 120588 = 1 the operator M120588 as in (8) is justthe Marcinkiewicz integral on (X 119889 120583) (see [22])
(2) If we take (X 119889 120583) = (R119899
|sdot| 120583) and 120582(119910 119905) fl 119905119899 then
the parameter 119892lowast
120581functionMlowast120588
120581as in (9) is just a parameter
Littlewood-Paley operatorwith nondoublingmeasures in [8]
The following definition of the atomic Hardy space wasintroduced by Htyonen et al (see [18])
Definition 8 (see [18]) Let 120577 isin (1infin) and 119901 isin (1infin] Afunction 119887 isin 119871
1
loc(120583) is called a (119901 1)120592-atomic block if
(a) there exists a ball 119861 such that supp 119887 sub 119861
(b) intX119887(119909)d120583(119909) = 0
(c) for any 119894 isin 1 2 there exist a function 119886119894supported
on ball 119861119894sub 119861 and a number 120592
119894isin C such that
119887 = 12059211198861+ 120592
21198862
1003817100381710038171003817119886119894
1003817100381710038171003817119871119901(120583)
le [120583 (120577119861119894)]
1119901minus1
119870minus1
119861119894 119861
(10)
Moreover let |119887|119867
1119901
atb (120583)fl |120592
1| + |120592
2|
We say a function 119891 isin 1198711
(120583) belongs to the atomicHardy space 119867
1119901
atb (120583) if there are atomic blocks 119887119894infin
119894=1such
that 119891 = suminfin
119894=1119887119894with sum
infin
119894=1|119887
119894|119867
1119901
atb (120583)lt infin The 1198671119901
atb (120583) normof 119891 is denoted by 119891
119867
1119901
atb (120583)= infsuminfin
119894=1|119887
119894|119867
1119901
atb (120583) where the
infimum is taken over all the possible decompositions of 119891 asabove
It was proved by Htyonen et al in [18] that the definitionof 1198671119901
atb (120583) is not related to the choice of 120577 and the spaces119867
1119901
atb (120583) and 1198671infin
atb (120583) have the same norms for 119901 isin (1infin]Thus for convenience we always denote1198671119901
atb (120583) by1198671
(120583)Nowwe give the Hormander-type condition on (X 119889 120583)
that is there exists a positive 119862 such that
sup119903gt0
119889(1199101199101015840)le119903
infin
sum
119894=1
119894 int
6119894119903lt119889(119909119910)le6
119894+1119903
[
10038161003816100381610038161003816119870 (119909 119910) minus 119870 (119909 119910
1015840
)
10038161003816100381610038161003816+
10038161003816100381610038161003816119870 (119910 119909) minus 119870 (119910
1015840
119909)
10038161003816100381610038161003816]
d120583 (119909)
119889 (119909 119910)
le 119862 (11)
Notice this condition is slightly stronger than (7)Now let us state the main theorems which generalize and
improve the corresponding results in [8]
Theorem 9 Let119870(119909 119910) satisfy (6) and (7) and letMlowast120588
120581be as
in (9) with 120588 isin (0infin) and 120581 isin (1infin) ThenMlowast120588
120581is bounded
on 119871119901
(120583) for any 119901 isin [2infin)
Theorem 10 Let 119870(119909 119910) satisfy (6) and (11) and letMlowast120588
120581be
as in (9) with 120588 isin (12infin) and 120581 isin (1infin) Then Mlowast120588
120581is
bounded from 1198711
(120583) into weak 1198711
(120583) namely there exists apositive constant 119862 such that for any 120591 gt 0 and 119891 isin 119871
1
(120583)
120583 (119909 isin X Mlowast120588
120581(119891) (119909) gt 120591) le 119862
100381710038171003817100381711989110038171003817100381710038171198711(120583)
120591
(12)
Theorem 11 Let 119870(119909 119910) satisfy (6) and (11) and let Mlowast120588
120581be
as in (9) with 120588 gt 12 and 120581 gt 1 Suppose thatMlowast120588
120581is bounded
on 1198712
(120583) ThenMlowast120588
120581is bounded from119867
1
(120583) into 1198711
(120583)
Applying the Marcinkiewicz interpolation theorem andTheorems 9 and 10 it is easy to get the following result
Corollary 12 Under the assumption of Theorem 10 Mlowast120588
120581is
bounded on 119871119901
(120583) for 119901 isin (1 2)
The organization of this paper is as follows In Section 2wewill give somepreliminary lemmasTheproofs of themaintheorems will be given in Section 3 Throughout this paper119862 stands for a positive constant which is independent of themain parameters but it may be different from line to line Forany 119864 sub X we use 120594
119864to denote its characteristic function
2 Preliminary Lemmas
In this section we make some preliminary lemmas which areused in the proof of the main results Firstly we recall someproperties of119870
119861119878as in (4) (see [17])
Lemma 13 (see [17]) (1) For all balls 119861 sub 119877 sub 119878 it holds truethat 119870
119861119877le 119870
119861119878
(2) For any 120585 isin [1infin) there exists a positive constant 119862120585
such that for all balls 119861 sub 119878 with 119903119878le 120585119903
119861 119870
119861119878le 119862
120585
(3) For any 984858 isin (1infin) there exists a positive constant 119862984858
depending on 984858 such that for all balls 119861119870119861
119861
984858 le 119862984858
(4) There exists a positive constant 119888 such that for all balls119861 sub 119877 sub 119878 119870
119861119878le 119870
119861119877+ 119888119870
119877119878 In particular if 119861 and 119877 are
concentric then 119888 = 1(5) There exists a positive constant such that for all balls
119861 sub 119877 sub 119878 119870119861119877
le 119870119861119878 moreover if 119861 and 119877 are concentric
then 119870119877119878
le 119870119861119878
4 Journal of Function Spaces
To state the following lemmas let us give a known-result(see [19]) For 120578 isin (0infin) the maximal operator is defined bysetting that for all 119891 isin 119871
1
loc(120583) and 119909 isin X
119872(120578)119891 (119909) fl sup
119876ni119909119876doubling
1
120583 (120578119876)
int
119876
1003816100381610038161003816119891 (119910)
1003816100381610038161003816d120583 (119910) (13)
is bounded on 119871119901
(120583) provided that 119901 isin (1infin) and alsobounded from 119871
1
(120583) into 1198711infin
(120583)The following lemma is slightly changed from [8]
Lemma 14 Let 119870(119909 119910) satisfy (6) and (7) and 120578 isin (0infin)Assume that M120588 is as in (8) and Mlowast120588
120581is as in (9) with
120588 isin (0infin) and 120581 isin (1infin) Then for any nonnegative function120601 there exists a positive constant 119862 such that for all 119891 isin 119871
119901
(120583)
with 119901 isin (1infin)
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
le 119862int
X
[M120588
(119891) (119909)]
2
119872120578(120601) (119909) d120583 (119909)
(14)
Proof By the definition ofMlowast120588
120581(119891) we have
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
= int
X
∬
Xtimes(0infin)
(
119905
119905 + 119889 (119909 119910)
)
120573 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588119891 (119910) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910)
120582 (119910 119905)
d119905119905
120601 (119909) d120583 (119909)
le int
X
int
infin
0
1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119910) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d119905119905
sup119905gt0
[int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)] d120583 (119910)
= int
X
[M120588
(119891) (119910)]
2 sup119905gt0
[int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)] d120583 (119910)
(15)
Thus to prove Lemma 14 we only need to estimate that
sup119905gt0
int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
le 119862119872120578(120601) (119910)
(16)
For any 119910 isin X and 119905 gt 0 write
int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
= int
119861(119910119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
+ int
X119861(119910119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
š 1198631+ 119863
2
(17)
For1198631 it is not difficult to obtain that
1198631le int
119861(119910119905)
120601 (119909)
120582 (119910 119905)
d120583 (119909)
=
120583 (120578119861 (119910 119905))
120582 (119910 119905)
1
120583 (120578119861 (119910 119905))
int
119861(119910119905)
120601 (119909) d120583 (119909)
le 119862119872120578(120601) (119910)
(18)
Now we turn to estimate1198632 by (2) and (13) we have
1198632le
infin
sum
119896=1
int
119861(1199106119896119905)119861(1199106
119896minus1119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
sdot
120601 (119909)
120582 (119910 119905)
d120583 (119909) le 119862
infin
sum
119896=1
6minus(119896minus1)120573
sdot int
119861(1199106119896119905)
120601 (119909)
120582 (119910 119905)
d120583 (119909) le 119862
infin
sum
119896=1
6minus(119896minus1)120573
sdot
120583 (119861 (119910 6119896
119905))
120582 (119910 119905)
1
120583 (119861 (119910 6119896119905))
int
119861(1199106119896119905)
120601 (119909) d120583 (119909)
le 119862
infin
sum
119896=1
6minus(119896minus1)120573
120583 (119861 (119910 6119896
119905))
120582 (119910 119905)
119872120578(120601) (119910) le 119862
sdot
120582 (119910 6119896
119905)
120582 (119910 119905)
119872120578(120601) (119910)
infin
sum
119896=1
6minus(119896minus1)120573
120582 (119910 6119896
119905)
120582 (119910 119905)
le 119862
sdot
120582 (119910 6119896
119905)
120582 (119910 119905)
119872120578(120601) (119910)
infin
sum
119896=1
6minus(119896minus1)120573
le 119862119872120578(120601) (119910)
(19)
Combining the estimates for 1198631and 119863
2 we obtain (16) and
hence complete the proof of Lemma 14
Finally we recall the Calderon-Zygmund decompositiontheorem (see [21]) Suppose that 120574
0is a fixed positive constant
Journal of Function Spaces 5
satisfying that 1205740gt max1198623 log
26
120582 6
3119899
where 119862120582is as in (2)
and 119899 as in Remark 3
Lemma 15 (see [21]) Let 119901 isin [1infin) 119891 isin 119871119901
(120583) and 119905 isin
(0infin) (119905 gt 1205740119891
119871119901(120583)
120583(X) when 120583(X) lt infin) Then
(1) there exists a family of finite overlapping balls 6119861119894119894
such that 119861119894119894is pairwise disjoint
1
120583 (62119861119894)
int
119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119889120583 (119909) gt
119905119901
1205740
forall119894 (20)
1
120583 (62120591119861
119894)
int
120591119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119889120583 (119909) le
119905119901
1205740
forall119894 forall120591 isin (2infin)
(21)
1003816100381610038161003816119891 (119909)
1003816100381610038161003816le 119905
119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X (⋃
119894
6119861119894)
(22)
(2) for each 119894 let 119878119894be a (3 times 6
2
119862
log2(3times62)+1
120582)-doubling ball
of the family (3times62
)119896
119861119894119896isinN and 120596119894
= 1205946119861119894
(sum1198961205946119861119896
)Then there exists a family 120593
119894119894of functions that for
each 119894 supp(120593119894) sub 119878
119894 120593
119894has a constant sign on 119878
119894and
int
X
120593119894(119909) 119889120583 (119909) = int
6119861119894
119891 (119909) 120596119894(119909) 119889120583 (119909)
sum
119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816le 120574119905 119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X
(23)
where 120574 is some positive constant depending only on (X 120583)and there exists a positive constant 119862 independent of 119891 119905 and119894 such that if 119901 = 1 then
1003817100381710038171003817120593119894
1003817100381710038171003817119871infin(120583)
120583 (119878119894) le 119862int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816119889120583 (119909) (24)
and if 119901 isin (1infin)
(int
119878119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816
119901
119889120583 (119909))
1119901
[120583 (119878119894)]
11199011015840
le
119862
119905119901minus1
int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816
119901
119889120583 (119909)
(25)
3 Proofs of Theorems
Proof ofTheorem 9 For the case of 119901 = 2 assume 120601(119909) = 1 inLemma 14 then it is easy to get that
int
X
[Mlowast120588
120581(119891) (119909)]
2 d120583 (119909)
le 119862int
X
[M120588
(119891) (119909)]
2 d120583 (119909)
(26)
which along with 1198712
(120583)-boundedness of M120588 easily yieldsthat Theorem 9 holds
For the case of 119901 gt 2 let 119902 be the index conjugate to1199012 By applying Holder inequality and Lemma 14 we canconclude
1003817100381710038171003817M
lowast120588
120581(119891)
1003817100381710038171003817
2
119871119901(120583)
= sup120601ge0
120601119871119902(120583)
le1
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
le 119862 sup120601ge0
120601119871119902(120583)
le1
int
X
[M120588
(119891) (119909)]
2
119872120578120601 (119909) d120583 (119909)
le 1198621003817100381710038171003817M
120588
(119891)1003817100381710038171003817
2
119871119901(120583)
sup120601ge0
120601119871119902(120583)
le1
10038171003817100381710038171003817119872
120578(120601)
10038171003817100381710038171003817119871119902(120583)
le 11986210038171003817100381710038171198911003817100381710038171003817
2
119871119901(120583)
sup120601ge0
120601119871119902(120583)
le1
10038171003817100381710038171206011003817100381710038171003817119871119902(120583)
le 11986210038171003817100381710038171198911003817100381710038171003817
2
119871119901(120583)
(27)
which is desired Thus we complete the proof of Theorem 9
Proof of Theorem 10 Without loss of generality we mayassume that 119891
1198711(120583)
= 1 It is easy to see that the conclusionofTheorem 10 naturally holds if 120591 le 120573
6(119891
1198711(120583)
120583(X)) when120583(X) lt infin Thus we only need to discuss the case that120591 gt 120573
6(119891
1198711(120583)
120583(X)) Applying Lemma 15 to119891 at the level 120591and letting120596
119894120593
119894119861
119894 and 119878
119894be the same as in Lemma 15 we see
that119891(119909) = 119887(119909)+ℎ(119909) where 119887(119909) fl 119891120594X⋃1198946119861119894
(119909)+sum119894120593119894(119909)
and ℎ(119909) fl sum119894[120596
119894(119909)119891(119909) minus 120593
119894(119909)] š sum
119894ℎ119894(119909) It is easy
to obtain that 119887119871infin(120583)
le 119862120591 and 1198871198711(120583)
le 119862 By 1198712
(120583)-boundedness ofMlowast120588
120581 we have
120583 (119909 isin X Mlowast120588
120581(119887) (119909) gt 120591) le
1003817100381710038171003817Mlowast120588
120581(119887)
1003817100381710038171003817
2
1198712(120583)
1205912
le 119862
1198872
1198712(120583)
1205912
le 119862120591minus1
(28)
On the other hand by (20) with 119901 = 1 and the fact that thesequence of balls 119861
119894119894 is pairwise disjoint we see that
120583(⋃
119894
62
119861119894) le 119862120591
minus1
int
X
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) le 119862120591
minus1
(29)
and thus the proof of theTheorem 10 can be reduced to provethat
120583(119909 isin X ⋃
119894
62
119861119894 M
lowast120588
120581(ℎ) (119909) gt 120591) le 119862120591
minus1
(30)
For each fixed 119894 denote the center of 119861119894by 119909
119894 and let 119873
1
be the positive integer satisfying 119878119894= (3 times 6
2
)1198731119861119894 We have
120583(119909 isin X ⋃
119894
62
119861119894 M
lowast120588
120581(ℎ) (119909) gt 120591)
le 120591minus1
sum
119894
int
X⋃11989462119861119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
6 Journal of Function Spaces
le 120591minus1
sum
119894
int
X6119878119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
+ 120591minus1
sum
119894
int
611987811989462119861119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
š 120591minus1
sum
119894
(1198641+ 119864
2)
(31)
Firstly let us estimate 1198642and write it as
1198642le int
611987811989462119861119894
Mlowast120588
120581(119891120596
119894) (119909) d120583 (119909)
+ int
611987811989462119861119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909) š 119864
21+ 119864
22
(32)
where ℎ119894fl 120596
119894119891 minus 120593
119894 By Holder inequality (24) and 119871
2
(120583)-boundedness ofMlowast120588
120581 we have
11986422
le int
6119878119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909)
le (int
6119878119894
1003816100381610038161003816M
lowast120588
120581(120593
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862(int
6119878119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816d120583 (119909)
(33)
For 11986421 by Minkowski inequality and (6) write
11986421
= int
611987811989462119861119894
[
[
∬
Xtimes(0infin)
1003816100381610038161003816100381610038161003816100381610038161003816
(
119905
119905 + 119889 (119909 119910)
)
1205812
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119911) 120596119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
le 119862int
611987811989462119861119894
int
X
1003816100381610038161003816119891 (119911) 120596
119894(119911)
1003816100381610038161003816[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905
2119889(119910119911)gt119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)lt119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
š 1198651+ 119865
2+ 119865
3
(34)
To this end let 119861119894be as in Lemma 15 with 119888
119861119894and 119903
119861119894being
respectively its center and radius For any 119909 isin 6119878119894 6
2
119861119894and
119911 isin 6119861119894 by (2) and (3) we have
1198651le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
int
infin
119889(119910119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) dt120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119911))
(int
infin
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119910 119911))]
3
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
Journal of Function Spaces 7
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
d120583 (119910)
[120582 (119910 (12) 119889 (119909 119911))]
2
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 (12) 119889 (119909 119911))]2int
2119889(119910119911)gt119889(119909119911)
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))119861(1199112119896minus2
119889(119909119911))
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
1
120582 (119911 119889 (119909 119911))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))
d120583 (119910)
120582 (119910 2119896minus2
119889 (119909 119911))
]
12
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(35)
where we use the fact that
int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) le 119862119870119861119894 119878119894
(36)
Next we estimate 1198652 For any 119909 isin 6119878
119894 6
2
119861119894 119910 isin X
and 119911 isin 6119861119894satisfying 119889(119910 119909) lt 119905 2119889(119910 119911) le 119889(119909 119911) and
(12)119889(119909 119911) lt 119905 we have
1198652le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
int
infin
(12)119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
d120583 (119910)
120583 (119861 (119910 119889 (119909 119911)))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(37)
Finally for any 119909 isin 6119878119894 6
2
119861119894 119910 isin X and 119911 isin 6119861
119894sat-
isfying 2119889(119910 119911) le 119889(119909 119911) 2119889(119910 119911) ge 119889(119909 119911) and 119889(119909 119910) lt
(32)119889(119909 119911) by applying (2) we have
1198653le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
1
120582 (119910 119889 (119909 119911))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 119889 (119909 119911))]2
120583 (119861 (119911 (12) 119889 (119909 119911)))
120582 (119911 119889 (119909 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(38)
8 Journal of Function Spaces
Combining the estimates for 1198651 119865
2 and 119865
3 we obtain that
11986421
le 119862int6119861119894
|119891(119911)|d120583(119911) where together with the fact that119864
22le 119862int
6119861119894
|119891(119911)|d120583(119911) we have
1198642le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (39)
Now we turn to estimate for 1198641 Let 119876
119894= 119861(119888
119861119894 119903
119878119894) and
write
1198641le int
X6119878119894
[∬
119889(119909119910)lt119905
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isin119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isinX119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 11986411
+ 11986412
+ 11986413
(40)
For each fixed 119894 decompose 11986411as
11986411
le int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isin2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isinX2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 1198681+ 119868
2
(41)
For any 119909 isin X 6119878119894 119910 isin 2119878
119894with 119889(119910 119909) lt 119905 and
119911 isin 119878119894 119889(119909 119888
119861119894) minus 2119903
119878119894le 119889(119909 119910) lt 119905 and 119889(119910 119911) lt 3119903
119878119894
togetherwithMinkowski inequality and (6) we can conclude
1198681le 119862int
X6119878119894
int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
∬119889(119909119910)lt119905119889(119910119911)le119905
119910isin2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
(int
infin
119889(119909119888119861119894)minus2119903119878119894
d119905120582 (119910 119905) 119905
1+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
1
[119889 (119909 119888119861119894) minus 2119903
119878119894]
2120588
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(42)
Journal of Function Spaces 9
For 1198682 write
1198682le int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905le119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
š 11986821
+ 11986822
(43)
For 11986821 by Minkowski inequality and (6) we deduce
11986821
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
119889(119910119888119861119894)+119903119878119894
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[int
2119896+1
611987811989421198966119878119894
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(44)
Nowwe estimate 11986822 ApplyingMinkowski inequality and the
vanishing moment we have
11986822
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870(119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588+
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
+ 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909) š 1198691+ 119869
2
(45)
With a way similar to that used in the proof of 1198681 we have
1198691le 119862ℎ
1198941198711(120583) Thus we only need to estimate 119869
2 by Mink-
owski inequality and (11) it follows that
1198692le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2minus2120588
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
infin
119889(119910119888119861119894)+119903119878119894
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
10 Journal of Function Spaces
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[
[
int
2119896119903119878119894
lt119889(1199101198881198611)le2119896+1
119903119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2d120583 (119910)
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(46)
Combining the estimates for 1198691 119869
2 119868
21 and 119868
1 we obtain that
11986411
le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(47)
Next we estimate 11986412 For any 119910 isin 119861
119894 119909 isin X 6119878
119894 and
119911 isin 119878119894 we have 119889(119909 119910) ge (12)119889(119909 119888
119861119894) 119889(119910 119911) le 2119903
119878119894 and
119889(119909 119910) sim 119889(119909 119888119861119894) and together with this fact Minkowski
inequality and (6) we get
11986412
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
[
∬119889(119909119910)ge119905
119889(119910119911)le119905
119910isin119876119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(48)
It remains to estimate 11986413 Applying Minkowski inequality
and (6) we have
11986413
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
+ 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)gt2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
š 1198801+ 119880
2
(49)
Journal of Function Spaces 11
Now we estimate 1198801 For any 119910 isin X 119876
119894 119911 isin 119878
119894 and 119889(119910
119911) le 119905 le 119889(119909 119911) it is easy to see 119889(119910 119911) sim 119889(119910 119888119861119894) So we have
1198801le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
d120583 (119910)
120582 (119910 119889 (119910 119888119861119894))
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(50)
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical PhysicsAdvances in
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 3
where 120588 isin (0infin)The parameter 119892lowast
120581functionMlowast120588
120581is defined
by
Mlowast120588
120581(119891) (119909) = ∬
Xtimes(0infin)
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
sdot int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910)
120582 (119910 119905)
d119905119905
12
(9)
where 119909 isin X X times (0infin) fl (119910 119905) 119910 isin X 119905 gt 0 120588 gt 0
and 120581 isin (1infin)
Remark 7 (1) When 120588 = 1 the operator M120588 as in (8) is justthe Marcinkiewicz integral on (X 119889 120583) (see [22])
(2) If we take (X 119889 120583) = (R119899
|sdot| 120583) and 120582(119910 119905) fl 119905119899 then
the parameter 119892lowast
120581functionMlowast120588
120581as in (9) is just a parameter
Littlewood-Paley operatorwith nondoublingmeasures in [8]
The following definition of the atomic Hardy space wasintroduced by Htyonen et al (see [18])
Definition 8 (see [18]) Let 120577 isin (1infin) and 119901 isin (1infin] Afunction 119887 isin 119871
1
loc(120583) is called a (119901 1)120592-atomic block if
(a) there exists a ball 119861 such that supp 119887 sub 119861
(b) intX119887(119909)d120583(119909) = 0
(c) for any 119894 isin 1 2 there exist a function 119886119894supported
on ball 119861119894sub 119861 and a number 120592
119894isin C such that
119887 = 12059211198861+ 120592
21198862
1003817100381710038171003817119886119894
1003817100381710038171003817119871119901(120583)
le [120583 (120577119861119894)]
1119901minus1
119870minus1
119861119894 119861
(10)
Moreover let |119887|119867
1119901
atb (120583)fl |120592
1| + |120592
2|
We say a function 119891 isin 1198711
(120583) belongs to the atomicHardy space 119867
1119901
atb (120583) if there are atomic blocks 119887119894infin
119894=1such
that 119891 = suminfin
119894=1119887119894with sum
infin
119894=1|119887
119894|119867
1119901
atb (120583)lt infin The 1198671119901
atb (120583) normof 119891 is denoted by 119891
119867
1119901
atb (120583)= infsuminfin
119894=1|119887
119894|119867
1119901
atb (120583) where the
infimum is taken over all the possible decompositions of 119891 asabove
It was proved by Htyonen et al in [18] that the definitionof 1198671119901
atb (120583) is not related to the choice of 120577 and the spaces119867
1119901
atb (120583) and 1198671infin
atb (120583) have the same norms for 119901 isin (1infin]Thus for convenience we always denote1198671119901
atb (120583) by1198671
(120583)Nowwe give the Hormander-type condition on (X 119889 120583)
that is there exists a positive 119862 such that
sup119903gt0
119889(1199101199101015840)le119903
infin
sum
119894=1
119894 int
6119894119903lt119889(119909119910)le6
119894+1119903
[
10038161003816100381610038161003816119870 (119909 119910) minus 119870 (119909 119910
1015840
)
10038161003816100381610038161003816+
10038161003816100381610038161003816119870 (119910 119909) minus 119870 (119910
1015840
119909)
10038161003816100381610038161003816]
d120583 (119909)
119889 (119909 119910)
le 119862 (11)
Notice this condition is slightly stronger than (7)Now let us state the main theorems which generalize and
improve the corresponding results in [8]
Theorem 9 Let119870(119909 119910) satisfy (6) and (7) and letMlowast120588
120581be as
in (9) with 120588 isin (0infin) and 120581 isin (1infin) ThenMlowast120588
120581is bounded
on 119871119901
(120583) for any 119901 isin [2infin)
Theorem 10 Let 119870(119909 119910) satisfy (6) and (11) and letMlowast120588
120581be
as in (9) with 120588 isin (12infin) and 120581 isin (1infin) Then Mlowast120588
120581is
bounded from 1198711
(120583) into weak 1198711
(120583) namely there exists apositive constant 119862 such that for any 120591 gt 0 and 119891 isin 119871
1
(120583)
120583 (119909 isin X Mlowast120588
120581(119891) (119909) gt 120591) le 119862
100381710038171003817100381711989110038171003817100381710038171198711(120583)
120591
(12)
Theorem 11 Let 119870(119909 119910) satisfy (6) and (11) and let Mlowast120588
120581be
as in (9) with 120588 gt 12 and 120581 gt 1 Suppose thatMlowast120588
120581is bounded
on 1198712
(120583) ThenMlowast120588
120581is bounded from119867
1
(120583) into 1198711
(120583)
Applying the Marcinkiewicz interpolation theorem andTheorems 9 and 10 it is easy to get the following result
Corollary 12 Under the assumption of Theorem 10 Mlowast120588
120581is
bounded on 119871119901
(120583) for 119901 isin (1 2)
The organization of this paper is as follows In Section 2wewill give somepreliminary lemmasTheproofs of themaintheorems will be given in Section 3 Throughout this paper119862 stands for a positive constant which is independent of themain parameters but it may be different from line to line Forany 119864 sub X we use 120594
119864to denote its characteristic function
2 Preliminary Lemmas
In this section we make some preliminary lemmas which areused in the proof of the main results Firstly we recall someproperties of119870
119861119878as in (4) (see [17])
Lemma 13 (see [17]) (1) For all balls 119861 sub 119877 sub 119878 it holds truethat 119870
119861119877le 119870
119861119878
(2) For any 120585 isin [1infin) there exists a positive constant 119862120585
such that for all balls 119861 sub 119878 with 119903119878le 120585119903
119861 119870
119861119878le 119862
120585
(3) For any 984858 isin (1infin) there exists a positive constant 119862984858
depending on 984858 such that for all balls 119861119870119861
119861
984858 le 119862984858
(4) There exists a positive constant 119888 such that for all balls119861 sub 119877 sub 119878 119870
119861119878le 119870
119861119877+ 119888119870
119877119878 In particular if 119861 and 119877 are
concentric then 119888 = 1(5) There exists a positive constant such that for all balls
119861 sub 119877 sub 119878 119870119861119877
le 119870119861119878 moreover if 119861 and 119877 are concentric
then 119870119877119878
le 119870119861119878
4 Journal of Function Spaces
To state the following lemmas let us give a known-result(see [19]) For 120578 isin (0infin) the maximal operator is defined bysetting that for all 119891 isin 119871
1
loc(120583) and 119909 isin X
119872(120578)119891 (119909) fl sup
119876ni119909119876doubling
1
120583 (120578119876)
int
119876
1003816100381610038161003816119891 (119910)
1003816100381610038161003816d120583 (119910) (13)
is bounded on 119871119901
(120583) provided that 119901 isin (1infin) and alsobounded from 119871
1
(120583) into 1198711infin
(120583)The following lemma is slightly changed from [8]
Lemma 14 Let 119870(119909 119910) satisfy (6) and (7) and 120578 isin (0infin)Assume that M120588 is as in (8) and Mlowast120588
120581is as in (9) with
120588 isin (0infin) and 120581 isin (1infin) Then for any nonnegative function120601 there exists a positive constant 119862 such that for all 119891 isin 119871
119901
(120583)
with 119901 isin (1infin)
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
le 119862int
X
[M120588
(119891) (119909)]
2
119872120578(120601) (119909) d120583 (119909)
(14)
Proof By the definition ofMlowast120588
120581(119891) we have
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
= int
X
∬
Xtimes(0infin)
(
119905
119905 + 119889 (119909 119910)
)
120573 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588119891 (119910) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910)
120582 (119910 119905)
d119905119905
120601 (119909) d120583 (119909)
le int
X
int
infin
0
1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119910) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d119905119905
sup119905gt0
[int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)] d120583 (119910)
= int
X
[M120588
(119891) (119910)]
2 sup119905gt0
[int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)] d120583 (119910)
(15)
Thus to prove Lemma 14 we only need to estimate that
sup119905gt0
int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
le 119862119872120578(120601) (119910)
(16)
For any 119910 isin X and 119905 gt 0 write
int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
= int
119861(119910119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
+ int
X119861(119910119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
š 1198631+ 119863
2
(17)
For1198631 it is not difficult to obtain that
1198631le int
119861(119910119905)
120601 (119909)
120582 (119910 119905)
d120583 (119909)
=
120583 (120578119861 (119910 119905))
120582 (119910 119905)
1
120583 (120578119861 (119910 119905))
int
119861(119910119905)
120601 (119909) d120583 (119909)
le 119862119872120578(120601) (119910)
(18)
Now we turn to estimate1198632 by (2) and (13) we have
1198632le
infin
sum
119896=1
int
119861(1199106119896119905)119861(1199106
119896minus1119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
sdot
120601 (119909)
120582 (119910 119905)
d120583 (119909) le 119862
infin
sum
119896=1
6minus(119896minus1)120573
sdot int
119861(1199106119896119905)
120601 (119909)
120582 (119910 119905)
d120583 (119909) le 119862
infin
sum
119896=1
6minus(119896minus1)120573
sdot
120583 (119861 (119910 6119896
119905))
120582 (119910 119905)
1
120583 (119861 (119910 6119896119905))
int
119861(1199106119896119905)
120601 (119909) d120583 (119909)
le 119862
infin
sum
119896=1
6minus(119896minus1)120573
120583 (119861 (119910 6119896
119905))
120582 (119910 119905)
119872120578(120601) (119910) le 119862
sdot
120582 (119910 6119896
119905)
120582 (119910 119905)
119872120578(120601) (119910)
infin
sum
119896=1
6minus(119896minus1)120573
120582 (119910 6119896
119905)
120582 (119910 119905)
le 119862
sdot
120582 (119910 6119896
119905)
120582 (119910 119905)
119872120578(120601) (119910)
infin
sum
119896=1
6minus(119896minus1)120573
le 119862119872120578(120601) (119910)
(19)
Combining the estimates for 1198631and 119863
2 we obtain (16) and
hence complete the proof of Lemma 14
Finally we recall the Calderon-Zygmund decompositiontheorem (see [21]) Suppose that 120574
0is a fixed positive constant
Journal of Function Spaces 5
satisfying that 1205740gt max1198623 log
26
120582 6
3119899
where 119862120582is as in (2)
and 119899 as in Remark 3
Lemma 15 (see [21]) Let 119901 isin [1infin) 119891 isin 119871119901
(120583) and 119905 isin
(0infin) (119905 gt 1205740119891
119871119901(120583)
120583(X) when 120583(X) lt infin) Then
(1) there exists a family of finite overlapping balls 6119861119894119894
such that 119861119894119894is pairwise disjoint
1
120583 (62119861119894)
int
119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119889120583 (119909) gt
119905119901
1205740
forall119894 (20)
1
120583 (62120591119861
119894)
int
120591119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119889120583 (119909) le
119905119901
1205740
forall119894 forall120591 isin (2infin)
(21)
1003816100381610038161003816119891 (119909)
1003816100381610038161003816le 119905
119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X (⋃
119894
6119861119894)
(22)
(2) for each 119894 let 119878119894be a (3 times 6
2
119862
log2(3times62)+1
120582)-doubling ball
of the family (3times62
)119896
119861119894119896isinN and 120596119894
= 1205946119861119894
(sum1198961205946119861119896
)Then there exists a family 120593
119894119894of functions that for
each 119894 supp(120593119894) sub 119878
119894 120593
119894has a constant sign on 119878
119894and
int
X
120593119894(119909) 119889120583 (119909) = int
6119861119894
119891 (119909) 120596119894(119909) 119889120583 (119909)
sum
119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816le 120574119905 119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X
(23)
where 120574 is some positive constant depending only on (X 120583)and there exists a positive constant 119862 independent of 119891 119905 and119894 such that if 119901 = 1 then
1003817100381710038171003817120593119894
1003817100381710038171003817119871infin(120583)
120583 (119878119894) le 119862int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816119889120583 (119909) (24)
and if 119901 isin (1infin)
(int
119878119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816
119901
119889120583 (119909))
1119901
[120583 (119878119894)]
11199011015840
le
119862
119905119901minus1
int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816
119901
119889120583 (119909)
(25)
3 Proofs of Theorems
Proof ofTheorem 9 For the case of 119901 = 2 assume 120601(119909) = 1 inLemma 14 then it is easy to get that
int
X
[Mlowast120588
120581(119891) (119909)]
2 d120583 (119909)
le 119862int
X
[M120588
(119891) (119909)]
2 d120583 (119909)
(26)
which along with 1198712
(120583)-boundedness of M120588 easily yieldsthat Theorem 9 holds
For the case of 119901 gt 2 let 119902 be the index conjugate to1199012 By applying Holder inequality and Lemma 14 we canconclude
1003817100381710038171003817M
lowast120588
120581(119891)
1003817100381710038171003817
2
119871119901(120583)
= sup120601ge0
120601119871119902(120583)
le1
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
le 119862 sup120601ge0
120601119871119902(120583)
le1
int
X
[M120588
(119891) (119909)]
2
119872120578120601 (119909) d120583 (119909)
le 1198621003817100381710038171003817M
120588
(119891)1003817100381710038171003817
2
119871119901(120583)
sup120601ge0
120601119871119902(120583)
le1
10038171003817100381710038171003817119872
120578(120601)
10038171003817100381710038171003817119871119902(120583)
le 11986210038171003817100381710038171198911003817100381710038171003817
2
119871119901(120583)
sup120601ge0
120601119871119902(120583)
le1
10038171003817100381710038171206011003817100381710038171003817119871119902(120583)
le 11986210038171003817100381710038171198911003817100381710038171003817
2
119871119901(120583)
(27)
which is desired Thus we complete the proof of Theorem 9
Proof of Theorem 10 Without loss of generality we mayassume that 119891
1198711(120583)
= 1 It is easy to see that the conclusionofTheorem 10 naturally holds if 120591 le 120573
6(119891
1198711(120583)
120583(X)) when120583(X) lt infin Thus we only need to discuss the case that120591 gt 120573
6(119891
1198711(120583)
120583(X)) Applying Lemma 15 to119891 at the level 120591and letting120596
119894120593
119894119861
119894 and 119878
119894be the same as in Lemma 15 we see
that119891(119909) = 119887(119909)+ℎ(119909) where 119887(119909) fl 119891120594X⋃1198946119861119894
(119909)+sum119894120593119894(119909)
and ℎ(119909) fl sum119894[120596
119894(119909)119891(119909) minus 120593
119894(119909)] š sum
119894ℎ119894(119909) It is easy
to obtain that 119887119871infin(120583)
le 119862120591 and 1198871198711(120583)
le 119862 By 1198712
(120583)-boundedness ofMlowast120588
120581 we have
120583 (119909 isin X Mlowast120588
120581(119887) (119909) gt 120591) le
1003817100381710038171003817Mlowast120588
120581(119887)
1003817100381710038171003817
2
1198712(120583)
1205912
le 119862
1198872
1198712(120583)
1205912
le 119862120591minus1
(28)
On the other hand by (20) with 119901 = 1 and the fact that thesequence of balls 119861
119894119894 is pairwise disjoint we see that
120583(⋃
119894
62
119861119894) le 119862120591
minus1
int
X
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) le 119862120591
minus1
(29)
and thus the proof of theTheorem 10 can be reduced to provethat
120583(119909 isin X ⋃
119894
62
119861119894 M
lowast120588
120581(ℎ) (119909) gt 120591) le 119862120591
minus1
(30)
For each fixed 119894 denote the center of 119861119894by 119909
119894 and let 119873
1
be the positive integer satisfying 119878119894= (3 times 6
2
)1198731119861119894 We have
120583(119909 isin X ⋃
119894
62
119861119894 M
lowast120588
120581(ℎ) (119909) gt 120591)
le 120591minus1
sum
119894
int
X⋃11989462119861119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
6 Journal of Function Spaces
le 120591minus1
sum
119894
int
X6119878119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
+ 120591minus1
sum
119894
int
611987811989462119861119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
š 120591minus1
sum
119894
(1198641+ 119864
2)
(31)
Firstly let us estimate 1198642and write it as
1198642le int
611987811989462119861119894
Mlowast120588
120581(119891120596
119894) (119909) d120583 (119909)
+ int
611987811989462119861119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909) š 119864
21+ 119864
22
(32)
where ℎ119894fl 120596
119894119891 minus 120593
119894 By Holder inequality (24) and 119871
2
(120583)-boundedness ofMlowast120588
120581 we have
11986422
le int
6119878119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909)
le (int
6119878119894
1003816100381610038161003816M
lowast120588
120581(120593
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862(int
6119878119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816d120583 (119909)
(33)
For 11986421 by Minkowski inequality and (6) write
11986421
= int
611987811989462119861119894
[
[
∬
Xtimes(0infin)
1003816100381610038161003816100381610038161003816100381610038161003816
(
119905
119905 + 119889 (119909 119910)
)
1205812
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119911) 120596119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
le 119862int
611987811989462119861119894
int
X
1003816100381610038161003816119891 (119911) 120596
119894(119911)
1003816100381610038161003816[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905
2119889(119910119911)gt119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)lt119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
š 1198651+ 119865
2+ 119865
3
(34)
To this end let 119861119894be as in Lemma 15 with 119888
119861119894and 119903
119861119894being
respectively its center and radius For any 119909 isin 6119878119894 6
2
119861119894and
119911 isin 6119861119894 by (2) and (3) we have
1198651le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
int
infin
119889(119910119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) dt120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119911))
(int
infin
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119910 119911))]
3
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
Journal of Function Spaces 7
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
d120583 (119910)
[120582 (119910 (12) 119889 (119909 119911))]
2
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 (12) 119889 (119909 119911))]2int
2119889(119910119911)gt119889(119909119911)
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))119861(1199112119896minus2
119889(119909119911))
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
1
120582 (119911 119889 (119909 119911))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))
d120583 (119910)
120582 (119910 2119896minus2
119889 (119909 119911))
]
12
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(35)
where we use the fact that
int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) le 119862119870119861119894 119878119894
(36)
Next we estimate 1198652 For any 119909 isin 6119878
119894 6
2
119861119894 119910 isin X
and 119911 isin 6119861119894satisfying 119889(119910 119909) lt 119905 2119889(119910 119911) le 119889(119909 119911) and
(12)119889(119909 119911) lt 119905 we have
1198652le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
int
infin
(12)119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
d120583 (119910)
120583 (119861 (119910 119889 (119909 119911)))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(37)
Finally for any 119909 isin 6119878119894 6
2
119861119894 119910 isin X and 119911 isin 6119861
119894sat-
isfying 2119889(119910 119911) le 119889(119909 119911) 2119889(119910 119911) ge 119889(119909 119911) and 119889(119909 119910) lt
(32)119889(119909 119911) by applying (2) we have
1198653le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
1
120582 (119910 119889 (119909 119911))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 119889 (119909 119911))]2
120583 (119861 (119911 (12) 119889 (119909 119911)))
120582 (119911 119889 (119909 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(38)
8 Journal of Function Spaces
Combining the estimates for 1198651 119865
2 and 119865
3 we obtain that
11986421
le 119862int6119861119894
|119891(119911)|d120583(119911) where together with the fact that119864
22le 119862int
6119861119894
|119891(119911)|d120583(119911) we have
1198642le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (39)
Now we turn to estimate for 1198641 Let 119876
119894= 119861(119888
119861119894 119903
119878119894) and
write
1198641le int
X6119878119894
[∬
119889(119909119910)lt119905
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isin119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isinX119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 11986411
+ 11986412
+ 11986413
(40)
For each fixed 119894 decompose 11986411as
11986411
le int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isin2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isinX2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 1198681+ 119868
2
(41)
For any 119909 isin X 6119878119894 119910 isin 2119878
119894with 119889(119910 119909) lt 119905 and
119911 isin 119878119894 119889(119909 119888
119861119894) minus 2119903
119878119894le 119889(119909 119910) lt 119905 and 119889(119910 119911) lt 3119903
119878119894
togetherwithMinkowski inequality and (6) we can conclude
1198681le 119862int
X6119878119894
int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
∬119889(119909119910)lt119905119889(119910119911)le119905
119910isin2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
(int
infin
119889(119909119888119861119894)minus2119903119878119894
d119905120582 (119910 119905) 119905
1+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
1
[119889 (119909 119888119861119894) minus 2119903
119878119894]
2120588
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(42)
Journal of Function Spaces 9
For 1198682 write
1198682le int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905le119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
š 11986821
+ 11986822
(43)
For 11986821 by Minkowski inequality and (6) we deduce
11986821
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
119889(119910119888119861119894)+119903119878119894
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[int
2119896+1
611987811989421198966119878119894
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(44)
Nowwe estimate 11986822 ApplyingMinkowski inequality and the
vanishing moment we have
11986822
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870(119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588+
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
+ 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909) š 1198691+ 119869
2
(45)
With a way similar to that used in the proof of 1198681 we have
1198691le 119862ℎ
1198941198711(120583) Thus we only need to estimate 119869
2 by Mink-
owski inequality and (11) it follows that
1198692le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2minus2120588
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
infin
119889(119910119888119861119894)+119903119878119894
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
10 Journal of Function Spaces
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[
[
int
2119896119903119878119894
lt119889(1199101198881198611)le2119896+1
119903119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2d120583 (119910)
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(46)
Combining the estimates for 1198691 119869
2 119868
21 and 119868
1 we obtain that
11986411
le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(47)
Next we estimate 11986412 For any 119910 isin 119861
119894 119909 isin X 6119878
119894 and
119911 isin 119878119894 we have 119889(119909 119910) ge (12)119889(119909 119888
119861119894) 119889(119910 119911) le 2119903
119878119894 and
119889(119909 119910) sim 119889(119909 119888119861119894) and together with this fact Minkowski
inequality and (6) we get
11986412
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
[
∬119889(119909119910)ge119905
119889(119910119911)le119905
119910isin119876119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(48)
It remains to estimate 11986413 Applying Minkowski inequality
and (6) we have
11986413
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
+ 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)gt2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
š 1198801+ 119880
2
(49)
Journal of Function Spaces 11
Now we estimate 1198801 For any 119910 isin X 119876
119894 119911 isin 119878
119894 and 119889(119910
119911) le 119905 le 119889(119909 119911) it is easy to see 119889(119910 119911) sim 119889(119910 119888119861119894) So we have
1198801le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
d120583 (119910)
120582 (119910 119889 (119910 119888119861119894))
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(50)
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Function Spaces
To state the following lemmas let us give a known-result(see [19]) For 120578 isin (0infin) the maximal operator is defined bysetting that for all 119891 isin 119871
1
loc(120583) and 119909 isin X
119872(120578)119891 (119909) fl sup
119876ni119909119876doubling
1
120583 (120578119876)
int
119876
1003816100381610038161003816119891 (119910)
1003816100381610038161003816d120583 (119910) (13)
is bounded on 119871119901
(120583) provided that 119901 isin (1infin) and alsobounded from 119871
1
(120583) into 1198711infin
(120583)The following lemma is slightly changed from [8]
Lemma 14 Let 119870(119909 119910) satisfy (6) and (7) and 120578 isin (0infin)Assume that M120588 is as in (8) and Mlowast120588
120581is as in (9) with
120588 isin (0infin) and 120581 isin (1infin) Then for any nonnegative function120601 there exists a positive constant 119862 such that for all 119891 isin 119871
119901
(120583)
with 119901 isin (1infin)
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
le 119862int
X
[M120588
(119891) (119909)]
2
119872120578(120601) (119909) d120583 (119909)
(14)
Proof By the definition ofMlowast120588
120581(119891) we have
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
= int
X
∬
Xtimes(0infin)
(
119905
119905 + 119889 (119909 119910)
)
120573 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588119891 (119910) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910)
120582 (119910 119905)
d119905119905
120601 (119909) d120583 (119909)
le int
X
int
infin
0
1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119910) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d119905119905
sup119905gt0
[int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)] d120583 (119910)
= int
X
[M120588
(119891) (119910)]
2 sup119905gt0
[int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)] d120583 (119910)
(15)
Thus to prove Lemma 14 we only need to estimate that
sup119905gt0
int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
le 119862119872120578(120601) (119910)
(16)
For any 119910 isin X and 119905 gt 0 write
int
X
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
= int
119861(119910119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
+ int
X119861(119910119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
120601 (119909)
120582 (119910 119905)
d120583 (119909)
š 1198631+ 119863
2
(17)
For1198631 it is not difficult to obtain that
1198631le int
119861(119910119905)
120601 (119909)
120582 (119910 119905)
d120583 (119909)
=
120583 (120578119861 (119910 119905))
120582 (119910 119905)
1
120583 (120578119861 (119910 119905))
int
119861(119910119905)
120601 (119909) d120583 (119909)
le 119862119872120578(120601) (119910)
(18)
Now we turn to estimate1198632 by (2) and (13) we have
1198632le
infin
sum
119896=1
int
119861(1199106119896119905)119861(1199106
119896minus1119905)
(
119905
119905 + 119889 (119909 119910)
)
120573
sdot
120601 (119909)
120582 (119910 119905)
d120583 (119909) le 119862
infin
sum
119896=1
6minus(119896minus1)120573
sdot int
119861(1199106119896119905)
120601 (119909)
120582 (119910 119905)
d120583 (119909) le 119862
infin
sum
119896=1
6minus(119896minus1)120573
sdot
120583 (119861 (119910 6119896
119905))
120582 (119910 119905)
1
120583 (119861 (119910 6119896119905))
int
119861(1199106119896119905)
120601 (119909) d120583 (119909)
le 119862
infin
sum
119896=1
6minus(119896minus1)120573
120583 (119861 (119910 6119896
119905))
120582 (119910 119905)
119872120578(120601) (119910) le 119862
sdot
120582 (119910 6119896
119905)
120582 (119910 119905)
119872120578(120601) (119910)
infin
sum
119896=1
6minus(119896minus1)120573
120582 (119910 6119896
119905)
120582 (119910 119905)
le 119862
sdot
120582 (119910 6119896
119905)
120582 (119910 119905)
119872120578(120601) (119910)
infin
sum
119896=1
6minus(119896minus1)120573
le 119862119872120578(120601) (119910)
(19)
Combining the estimates for 1198631and 119863
2 we obtain (16) and
hence complete the proof of Lemma 14
Finally we recall the Calderon-Zygmund decompositiontheorem (see [21]) Suppose that 120574
0is a fixed positive constant
Journal of Function Spaces 5
satisfying that 1205740gt max1198623 log
26
120582 6
3119899
where 119862120582is as in (2)
and 119899 as in Remark 3
Lemma 15 (see [21]) Let 119901 isin [1infin) 119891 isin 119871119901
(120583) and 119905 isin
(0infin) (119905 gt 1205740119891
119871119901(120583)
120583(X) when 120583(X) lt infin) Then
(1) there exists a family of finite overlapping balls 6119861119894119894
such that 119861119894119894is pairwise disjoint
1
120583 (62119861119894)
int
119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119889120583 (119909) gt
119905119901
1205740
forall119894 (20)
1
120583 (62120591119861
119894)
int
120591119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119889120583 (119909) le
119905119901
1205740
forall119894 forall120591 isin (2infin)
(21)
1003816100381610038161003816119891 (119909)
1003816100381610038161003816le 119905
119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X (⋃
119894
6119861119894)
(22)
(2) for each 119894 let 119878119894be a (3 times 6
2
119862
log2(3times62)+1
120582)-doubling ball
of the family (3times62
)119896
119861119894119896isinN and 120596119894
= 1205946119861119894
(sum1198961205946119861119896
)Then there exists a family 120593
119894119894of functions that for
each 119894 supp(120593119894) sub 119878
119894 120593
119894has a constant sign on 119878
119894and
int
X
120593119894(119909) 119889120583 (119909) = int
6119861119894
119891 (119909) 120596119894(119909) 119889120583 (119909)
sum
119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816le 120574119905 119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X
(23)
where 120574 is some positive constant depending only on (X 120583)and there exists a positive constant 119862 independent of 119891 119905 and119894 such that if 119901 = 1 then
1003817100381710038171003817120593119894
1003817100381710038171003817119871infin(120583)
120583 (119878119894) le 119862int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816119889120583 (119909) (24)
and if 119901 isin (1infin)
(int
119878119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816
119901
119889120583 (119909))
1119901
[120583 (119878119894)]
11199011015840
le
119862
119905119901minus1
int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816
119901
119889120583 (119909)
(25)
3 Proofs of Theorems
Proof ofTheorem 9 For the case of 119901 = 2 assume 120601(119909) = 1 inLemma 14 then it is easy to get that
int
X
[Mlowast120588
120581(119891) (119909)]
2 d120583 (119909)
le 119862int
X
[M120588
(119891) (119909)]
2 d120583 (119909)
(26)
which along with 1198712
(120583)-boundedness of M120588 easily yieldsthat Theorem 9 holds
For the case of 119901 gt 2 let 119902 be the index conjugate to1199012 By applying Holder inequality and Lemma 14 we canconclude
1003817100381710038171003817M
lowast120588
120581(119891)
1003817100381710038171003817
2
119871119901(120583)
= sup120601ge0
120601119871119902(120583)
le1
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
le 119862 sup120601ge0
120601119871119902(120583)
le1
int
X
[M120588
(119891) (119909)]
2
119872120578120601 (119909) d120583 (119909)
le 1198621003817100381710038171003817M
120588
(119891)1003817100381710038171003817
2
119871119901(120583)
sup120601ge0
120601119871119902(120583)
le1
10038171003817100381710038171003817119872
120578(120601)
10038171003817100381710038171003817119871119902(120583)
le 11986210038171003817100381710038171198911003817100381710038171003817
2
119871119901(120583)
sup120601ge0
120601119871119902(120583)
le1
10038171003817100381710038171206011003817100381710038171003817119871119902(120583)
le 11986210038171003817100381710038171198911003817100381710038171003817
2
119871119901(120583)
(27)
which is desired Thus we complete the proof of Theorem 9
Proof of Theorem 10 Without loss of generality we mayassume that 119891
1198711(120583)
= 1 It is easy to see that the conclusionofTheorem 10 naturally holds if 120591 le 120573
6(119891
1198711(120583)
120583(X)) when120583(X) lt infin Thus we only need to discuss the case that120591 gt 120573
6(119891
1198711(120583)
120583(X)) Applying Lemma 15 to119891 at the level 120591and letting120596
119894120593
119894119861
119894 and 119878
119894be the same as in Lemma 15 we see
that119891(119909) = 119887(119909)+ℎ(119909) where 119887(119909) fl 119891120594X⋃1198946119861119894
(119909)+sum119894120593119894(119909)
and ℎ(119909) fl sum119894[120596
119894(119909)119891(119909) minus 120593
119894(119909)] š sum
119894ℎ119894(119909) It is easy
to obtain that 119887119871infin(120583)
le 119862120591 and 1198871198711(120583)
le 119862 By 1198712
(120583)-boundedness ofMlowast120588
120581 we have
120583 (119909 isin X Mlowast120588
120581(119887) (119909) gt 120591) le
1003817100381710038171003817Mlowast120588
120581(119887)
1003817100381710038171003817
2
1198712(120583)
1205912
le 119862
1198872
1198712(120583)
1205912
le 119862120591minus1
(28)
On the other hand by (20) with 119901 = 1 and the fact that thesequence of balls 119861
119894119894 is pairwise disjoint we see that
120583(⋃
119894
62
119861119894) le 119862120591
minus1
int
X
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) le 119862120591
minus1
(29)
and thus the proof of theTheorem 10 can be reduced to provethat
120583(119909 isin X ⋃
119894
62
119861119894 M
lowast120588
120581(ℎ) (119909) gt 120591) le 119862120591
minus1
(30)
For each fixed 119894 denote the center of 119861119894by 119909
119894 and let 119873
1
be the positive integer satisfying 119878119894= (3 times 6
2
)1198731119861119894 We have
120583(119909 isin X ⋃
119894
62
119861119894 M
lowast120588
120581(ℎ) (119909) gt 120591)
le 120591minus1
sum
119894
int
X⋃11989462119861119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
6 Journal of Function Spaces
le 120591minus1
sum
119894
int
X6119878119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
+ 120591minus1
sum
119894
int
611987811989462119861119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
š 120591minus1
sum
119894
(1198641+ 119864
2)
(31)
Firstly let us estimate 1198642and write it as
1198642le int
611987811989462119861119894
Mlowast120588
120581(119891120596
119894) (119909) d120583 (119909)
+ int
611987811989462119861119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909) š 119864
21+ 119864
22
(32)
where ℎ119894fl 120596
119894119891 minus 120593
119894 By Holder inequality (24) and 119871
2
(120583)-boundedness ofMlowast120588
120581 we have
11986422
le int
6119878119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909)
le (int
6119878119894
1003816100381610038161003816M
lowast120588
120581(120593
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862(int
6119878119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816d120583 (119909)
(33)
For 11986421 by Minkowski inequality and (6) write
11986421
= int
611987811989462119861119894
[
[
∬
Xtimes(0infin)
1003816100381610038161003816100381610038161003816100381610038161003816
(
119905
119905 + 119889 (119909 119910)
)
1205812
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119911) 120596119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
le 119862int
611987811989462119861119894
int
X
1003816100381610038161003816119891 (119911) 120596
119894(119911)
1003816100381610038161003816[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905
2119889(119910119911)gt119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)lt119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
š 1198651+ 119865
2+ 119865
3
(34)
To this end let 119861119894be as in Lemma 15 with 119888
119861119894and 119903
119861119894being
respectively its center and radius For any 119909 isin 6119878119894 6
2
119861119894and
119911 isin 6119861119894 by (2) and (3) we have
1198651le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
int
infin
119889(119910119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) dt120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119911))
(int
infin
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119910 119911))]
3
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
Journal of Function Spaces 7
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
d120583 (119910)
[120582 (119910 (12) 119889 (119909 119911))]
2
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 (12) 119889 (119909 119911))]2int
2119889(119910119911)gt119889(119909119911)
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))119861(1199112119896minus2
119889(119909119911))
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
1
120582 (119911 119889 (119909 119911))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))
d120583 (119910)
120582 (119910 2119896minus2
119889 (119909 119911))
]
12
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(35)
where we use the fact that
int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) le 119862119870119861119894 119878119894
(36)
Next we estimate 1198652 For any 119909 isin 6119878
119894 6
2
119861119894 119910 isin X
and 119911 isin 6119861119894satisfying 119889(119910 119909) lt 119905 2119889(119910 119911) le 119889(119909 119911) and
(12)119889(119909 119911) lt 119905 we have
1198652le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
int
infin
(12)119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
d120583 (119910)
120583 (119861 (119910 119889 (119909 119911)))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(37)
Finally for any 119909 isin 6119878119894 6
2
119861119894 119910 isin X and 119911 isin 6119861
119894sat-
isfying 2119889(119910 119911) le 119889(119909 119911) 2119889(119910 119911) ge 119889(119909 119911) and 119889(119909 119910) lt
(32)119889(119909 119911) by applying (2) we have
1198653le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
1
120582 (119910 119889 (119909 119911))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 119889 (119909 119911))]2
120583 (119861 (119911 (12) 119889 (119909 119911)))
120582 (119911 119889 (119909 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(38)
8 Journal of Function Spaces
Combining the estimates for 1198651 119865
2 and 119865
3 we obtain that
11986421
le 119862int6119861119894
|119891(119911)|d120583(119911) where together with the fact that119864
22le 119862int
6119861119894
|119891(119911)|d120583(119911) we have
1198642le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (39)
Now we turn to estimate for 1198641 Let 119876
119894= 119861(119888
119861119894 119903
119878119894) and
write
1198641le int
X6119878119894
[∬
119889(119909119910)lt119905
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isin119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isinX119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 11986411
+ 11986412
+ 11986413
(40)
For each fixed 119894 decompose 11986411as
11986411
le int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isin2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isinX2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 1198681+ 119868
2
(41)
For any 119909 isin X 6119878119894 119910 isin 2119878
119894with 119889(119910 119909) lt 119905 and
119911 isin 119878119894 119889(119909 119888
119861119894) minus 2119903
119878119894le 119889(119909 119910) lt 119905 and 119889(119910 119911) lt 3119903
119878119894
togetherwithMinkowski inequality and (6) we can conclude
1198681le 119862int
X6119878119894
int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
∬119889(119909119910)lt119905119889(119910119911)le119905
119910isin2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
(int
infin
119889(119909119888119861119894)minus2119903119878119894
d119905120582 (119910 119905) 119905
1+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
1
[119889 (119909 119888119861119894) minus 2119903
119878119894]
2120588
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(42)
Journal of Function Spaces 9
For 1198682 write
1198682le int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905le119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
š 11986821
+ 11986822
(43)
For 11986821 by Minkowski inequality and (6) we deduce
11986821
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
119889(119910119888119861119894)+119903119878119894
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[int
2119896+1
611987811989421198966119878119894
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(44)
Nowwe estimate 11986822 ApplyingMinkowski inequality and the
vanishing moment we have
11986822
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870(119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588+
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
+ 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909) š 1198691+ 119869
2
(45)
With a way similar to that used in the proof of 1198681 we have
1198691le 119862ℎ
1198941198711(120583) Thus we only need to estimate 119869
2 by Mink-
owski inequality and (11) it follows that
1198692le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2minus2120588
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
infin
119889(119910119888119861119894)+119903119878119894
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
10 Journal of Function Spaces
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[
[
int
2119896119903119878119894
lt119889(1199101198881198611)le2119896+1
119903119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2d120583 (119910)
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(46)
Combining the estimates for 1198691 119869
2 119868
21 and 119868
1 we obtain that
11986411
le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(47)
Next we estimate 11986412 For any 119910 isin 119861
119894 119909 isin X 6119878
119894 and
119911 isin 119878119894 we have 119889(119909 119910) ge (12)119889(119909 119888
119861119894) 119889(119910 119911) le 2119903
119878119894 and
119889(119909 119910) sim 119889(119909 119888119861119894) and together with this fact Minkowski
inequality and (6) we get
11986412
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
[
∬119889(119909119910)ge119905
119889(119910119911)le119905
119910isin119876119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(48)
It remains to estimate 11986413 Applying Minkowski inequality
and (6) we have
11986413
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
+ 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)gt2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
š 1198801+ 119880
2
(49)
Journal of Function Spaces 11
Now we estimate 1198801 For any 119910 isin X 119876
119894 119911 isin 119878
119894 and 119889(119910
119911) le 119905 le 119889(119909 119911) it is easy to see 119889(119910 119911) sim 119889(119910 119888119861119894) So we have
1198801le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
d120583 (119910)
120582 (119910 119889 (119910 119888119861119894))
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(50)
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 5
satisfying that 1205740gt max1198623 log
26
120582 6
3119899
where 119862120582is as in (2)
and 119899 as in Remark 3
Lemma 15 (see [21]) Let 119901 isin [1infin) 119891 isin 119871119901
(120583) and 119905 isin
(0infin) (119905 gt 1205740119891
119871119901(120583)
120583(X) when 120583(X) lt infin) Then
(1) there exists a family of finite overlapping balls 6119861119894119894
such that 119861119894119894is pairwise disjoint
1
120583 (62119861119894)
int
119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119889120583 (119909) gt
119905119901
1205740
forall119894 (20)
1
120583 (62120591119861
119894)
int
120591119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816
119901
119889120583 (119909) le
119905119901
1205740
forall119894 forall120591 isin (2infin)
(21)
1003816100381610038161003816119891 (119909)
1003816100381610038161003816le 119905
119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X (⋃
119894
6119861119894)
(22)
(2) for each 119894 let 119878119894be a (3 times 6
2
119862
log2(3times62)+1
120582)-doubling ball
of the family (3times62
)119896
119861119894119896isinN and 120596119894
= 1205946119861119894
(sum1198961205946119861119896
)Then there exists a family 120593
119894119894of functions that for
each 119894 supp(120593119894) sub 119878
119894 120593
119894has a constant sign on 119878
119894and
int
X
120593119894(119909) 119889120583 (119909) = int
6119861119894
119891 (119909) 120596119894(119909) 119889120583 (119909)
sum
119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816le 120574119905 119891119900119903 120583-119886119897119898119900119904119905 119890V119890119903119910 119909 isin X
(23)
where 120574 is some positive constant depending only on (X 120583)and there exists a positive constant 119862 independent of 119891 119905 and119894 such that if 119901 = 1 then
1003817100381710038171003817120593119894
1003817100381710038171003817119871infin(120583)
120583 (119878119894) le 119862int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816119889120583 (119909) (24)
and if 119901 isin (1infin)
(int
119878119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816
119901
119889120583 (119909))
1119901
[120583 (119878119894)]
11199011015840
le
119862
119905119901minus1
int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816
119901
119889120583 (119909)
(25)
3 Proofs of Theorems
Proof ofTheorem 9 For the case of 119901 = 2 assume 120601(119909) = 1 inLemma 14 then it is easy to get that
int
X
[Mlowast120588
120581(119891) (119909)]
2 d120583 (119909)
le 119862int
X
[M120588
(119891) (119909)]
2 d120583 (119909)
(26)
which along with 1198712
(120583)-boundedness of M120588 easily yieldsthat Theorem 9 holds
For the case of 119901 gt 2 let 119902 be the index conjugate to1199012 By applying Holder inequality and Lemma 14 we canconclude
1003817100381710038171003817M
lowast120588
120581(119891)
1003817100381710038171003817
2
119871119901(120583)
= sup120601ge0
120601119871119902(120583)
le1
int
X
[Mlowast120588
120581(119891) (119909)]
2
120601 (119909) d120583 (119909)
le 119862 sup120601ge0
120601119871119902(120583)
le1
int
X
[M120588
(119891) (119909)]
2
119872120578120601 (119909) d120583 (119909)
le 1198621003817100381710038171003817M
120588
(119891)1003817100381710038171003817
2
119871119901(120583)
sup120601ge0
120601119871119902(120583)
le1
10038171003817100381710038171003817119872
120578(120601)
10038171003817100381710038171003817119871119902(120583)
le 11986210038171003817100381710038171198911003817100381710038171003817
2
119871119901(120583)
sup120601ge0
120601119871119902(120583)
le1
10038171003817100381710038171206011003817100381710038171003817119871119902(120583)
le 11986210038171003817100381710038171198911003817100381710038171003817
2
119871119901(120583)
(27)
which is desired Thus we complete the proof of Theorem 9
Proof of Theorem 10 Without loss of generality we mayassume that 119891
1198711(120583)
= 1 It is easy to see that the conclusionofTheorem 10 naturally holds if 120591 le 120573
6(119891
1198711(120583)
120583(X)) when120583(X) lt infin Thus we only need to discuss the case that120591 gt 120573
6(119891
1198711(120583)
120583(X)) Applying Lemma 15 to119891 at the level 120591and letting120596
119894120593
119894119861
119894 and 119878
119894be the same as in Lemma 15 we see
that119891(119909) = 119887(119909)+ℎ(119909) where 119887(119909) fl 119891120594X⋃1198946119861119894
(119909)+sum119894120593119894(119909)
and ℎ(119909) fl sum119894[120596
119894(119909)119891(119909) minus 120593
119894(119909)] š sum
119894ℎ119894(119909) It is easy
to obtain that 119887119871infin(120583)
le 119862120591 and 1198871198711(120583)
le 119862 By 1198712
(120583)-boundedness ofMlowast120588
120581 we have
120583 (119909 isin X Mlowast120588
120581(119887) (119909) gt 120591) le
1003817100381710038171003817Mlowast120588
120581(119887)
1003817100381710038171003817
2
1198712(120583)
1205912
le 119862
1198872
1198712(120583)
1205912
le 119862120591minus1
(28)
On the other hand by (20) with 119901 = 1 and the fact that thesequence of balls 119861
119894119894 is pairwise disjoint we see that
120583(⋃
119894
62
119861119894) le 119862120591
minus1
int
X
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) le 119862120591
minus1
(29)
and thus the proof of theTheorem 10 can be reduced to provethat
120583(119909 isin X ⋃
119894
62
119861119894 M
lowast120588
120581(ℎ) (119909) gt 120591) le 119862120591
minus1
(30)
For each fixed 119894 denote the center of 119861119894by 119909
119894 and let 119873
1
be the positive integer satisfying 119878119894= (3 times 6
2
)1198731119861119894 We have
120583(119909 isin X ⋃
119894
62
119861119894 M
lowast120588
120581(ℎ) (119909) gt 120591)
le 120591minus1
sum
119894
int
X⋃11989462119861119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
6 Journal of Function Spaces
le 120591minus1
sum
119894
int
X6119878119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
+ 120591minus1
sum
119894
int
611987811989462119861119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
š 120591minus1
sum
119894
(1198641+ 119864
2)
(31)
Firstly let us estimate 1198642and write it as
1198642le int
611987811989462119861119894
Mlowast120588
120581(119891120596
119894) (119909) d120583 (119909)
+ int
611987811989462119861119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909) š 119864
21+ 119864
22
(32)
where ℎ119894fl 120596
119894119891 minus 120593
119894 By Holder inequality (24) and 119871
2
(120583)-boundedness ofMlowast120588
120581 we have
11986422
le int
6119878119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909)
le (int
6119878119894
1003816100381610038161003816M
lowast120588
120581(120593
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862(int
6119878119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816d120583 (119909)
(33)
For 11986421 by Minkowski inequality and (6) write
11986421
= int
611987811989462119861119894
[
[
∬
Xtimes(0infin)
1003816100381610038161003816100381610038161003816100381610038161003816
(
119905
119905 + 119889 (119909 119910)
)
1205812
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119911) 120596119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
le 119862int
611987811989462119861119894
int
X
1003816100381610038161003816119891 (119911) 120596
119894(119911)
1003816100381610038161003816[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905
2119889(119910119911)gt119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)lt119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
š 1198651+ 119865
2+ 119865
3
(34)
To this end let 119861119894be as in Lemma 15 with 119888
119861119894and 119903
119861119894being
respectively its center and radius For any 119909 isin 6119878119894 6
2
119861119894and
119911 isin 6119861119894 by (2) and (3) we have
1198651le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
int
infin
119889(119910119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) dt120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119911))
(int
infin
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119910 119911))]
3
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
Journal of Function Spaces 7
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
d120583 (119910)
[120582 (119910 (12) 119889 (119909 119911))]
2
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 (12) 119889 (119909 119911))]2int
2119889(119910119911)gt119889(119909119911)
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))119861(1199112119896minus2
119889(119909119911))
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
1
120582 (119911 119889 (119909 119911))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))
d120583 (119910)
120582 (119910 2119896minus2
119889 (119909 119911))
]
12
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(35)
where we use the fact that
int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) le 119862119870119861119894 119878119894
(36)
Next we estimate 1198652 For any 119909 isin 6119878
119894 6
2
119861119894 119910 isin X
and 119911 isin 6119861119894satisfying 119889(119910 119909) lt 119905 2119889(119910 119911) le 119889(119909 119911) and
(12)119889(119909 119911) lt 119905 we have
1198652le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
int
infin
(12)119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
d120583 (119910)
120583 (119861 (119910 119889 (119909 119911)))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(37)
Finally for any 119909 isin 6119878119894 6
2
119861119894 119910 isin X and 119911 isin 6119861
119894sat-
isfying 2119889(119910 119911) le 119889(119909 119911) 2119889(119910 119911) ge 119889(119909 119911) and 119889(119909 119910) lt
(32)119889(119909 119911) by applying (2) we have
1198653le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
1
120582 (119910 119889 (119909 119911))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 119889 (119909 119911))]2
120583 (119861 (119911 (12) 119889 (119909 119911)))
120582 (119911 119889 (119909 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(38)
8 Journal of Function Spaces
Combining the estimates for 1198651 119865
2 and 119865
3 we obtain that
11986421
le 119862int6119861119894
|119891(119911)|d120583(119911) where together with the fact that119864
22le 119862int
6119861119894
|119891(119911)|d120583(119911) we have
1198642le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (39)
Now we turn to estimate for 1198641 Let 119876
119894= 119861(119888
119861119894 119903
119878119894) and
write
1198641le int
X6119878119894
[∬
119889(119909119910)lt119905
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isin119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isinX119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 11986411
+ 11986412
+ 11986413
(40)
For each fixed 119894 decompose 11986411as
11986411
le int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isin2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isinX2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 1198681+ 119868
2
(41)
For any 119909 isin X 6119878119894 119910 isin 2119878
119894with 119889(119910 119909) lt 119905 and
119911 isin 119878119894 119889(119909 119888
119861119894) minus 2119903
119878119894le 119889(119909 119910) lt 119905 and 119889(119910 119911) lt 3119903
119878119894
togetherwithMinkowski inequality and (6) we can conclude
1198681le 119862int
X6119878119894
int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
∬119889(119909119910)lt119905119889(119910119911)le119905
119910isin2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
(int
infin
119889(119909119888119861119894)minus2119903119878119894
d119905120582 (119910 119905) 119905
1+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
1
[119889 (119909 119888119861119894) minus 2119903
119878119894]
2120588
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(42)
Journal of Function Spaces 9
For 1198682 write
1198682le int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905le119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
š 11986821
+ 11986822
(43)
For 11986821 by Minkowski inequality and (6) we deduce
11986821
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
119889(119910119888119861119894)+119903119878119894
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[int
2119896+1
611987811989421198966119878119894
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(44)
Nowwe estimate 11986822 ApplyingMinkowski inequality and the
vanishing moment we have
11986822
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870(119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588+
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
+ 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909) š 1198691+ 119869
2
(45)
With a way similar to that used in the proof of 1198681 we have
1198691le 119862ℎ
1198941198711(120583) Thus we only need to estimate 119869
2 by Mink-
owski inequality and (11) it follows that
1198692le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2minus2120588
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
infin
119889(119910119888119861119894)+119903119878119894
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
10 Journal of Function Spaces
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[
[
int
2119896119903119878119894
lt119889(1199101198881198611)le2119896+1
119903119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2d120583 (119910)
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(46)
Combining the estimates for 1198691 119869
2 119868
21 and 119868
1 we obtain that
11986411
le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(47)
Next we estimate 11986412 For any 119910 isin 119861
119894 119909 isin X 6119878
119894 and
119911 isin 119878119894 we have 119889(119909 119910) ge (12)119889(119909 119888
119861119894) 119889(119910 119911) le 2119903
119878119894 and
119889(119909 119910) sim 119889(119909 119888119861119894) and together with this fact Minkowski
inequality and (6) we get
11986412
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
[
∬119889(119909119910)ge119905
119889(119910119911)le119905
119910isin119876119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(48)
It remains to estimate 11986413 Applying Minkowski inequality
and (6) we have
11986413
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
+ 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)gt2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
š 1198801+ 119880
2
(49)
Journal of Function Spaces 11
Now we estimate 1198801 For any 119910 isin X 119876
119894 119911 isin 119878
119894 and 119889(119910
119911) le 119905 le 119889(119909 119911) it is easy to see 119889(119910 119911) sim 119889(119910 119888119861119894) So we have
1198801le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
d120583 (119910)
120582 (119910 119889 (119910 119888119861119894))
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(50)
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Function Spaces
le 120591minus1
sum
119894
int
X6119878119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
+ 120591minus1
sum
119894
int
611987811989462119861119894
Mlowast120588
120581(ℎ
119894) (119909) d120583 (119909)
š 120591minus1
sum
119894
(1198641+ 119864
2)
(31)
Firstly let us estimate 1198642and write it as
1198642le int
611987811989462119861119894
Mlowast120588
120581(119891120596
119894) (119909) d120583 (119909)
+ int
611987811989462119861119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909) š 119864
21+ 119864
22
(32)
where ℎ119894fl 120596
119894119891 minus 120593
119894 By Holder inequality (24) and 119871
2
(120583)-boundedness ofMlowast120588
120581 we have
11986422
le int
6119878119894
Mlowast120588
120581(120593
119894) (119909) d120583 (119909)
le (int
6119878119894
1003816100381610038161003816M
lowast120588
120581(120593
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862(int
6119878119894
1003816100381610038161003816120593119894(119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (6119878119894)
12
le 119862int
X
1003816100381610038161003816119891 (119909) 120596
119894(119909)
1003816100381610038161003816d120583 (119909)
(33)
For 11986421 by Minkowski inequality and (6) write
11986421
= int
611987811989462119861119894
[
[
∬
Xtimes(0infin)
1003816100381610038161003816100381610038161003816100381610038161003816
(
119905
119905 + 119889 (119909 119910)
)
1205812
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
119891 (119911) 120596119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
le 119862int
611987811989462119861119894
int
X
1003816100381610038161003816119891 (119911) 120596
119894(119911)
1003816100381610038161003816[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[∬
119889(119910119911)le119905
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905
2119889(119910119911)gt119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)lt119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
+ 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
š 1198651+ 119865
2+ 119865
3
(34)
To this end let 119861119894be as in Lemma 15 with 119888
119861119894and 119903
119861119894being
respectively its center and radius For any 119909 isin 6119878119894 6
2
119861119894and
119911 isin 6119861119894 by (2) and (3) we have
1198651le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
int
infin
119889(119910119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) dt120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119911))
(int
infin
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119910 119911))]
3
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
Journal of Function Spaces 7
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
d120583 (119910)
[120582 (119910 (12) 119889 (119909 119911))]
2
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 (12) 119889 (119909 119911))]2int
2119889(119910119911)gt119889(119909119911)
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))119861(1199112119896minus2
119889(119909119911))
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
1
120582 (119911 119889 (119909 119911))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))
d120583 (119910)
120582 (119910 2119896minus2
119889 (119909 119911))
]
12
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(35)
where we use the fact that
int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) le 119862119870119861119894 119878119894
(36)
Next we estimate 1198652 For any 119909 isin 6119878
119894 6
2
119861119894 119910 isin X
and 119911 isin 6119861119894satisfying 119889(119910 119909) lt 119905 2119889(119910 119911) le 119889(119909 119911) and
(12)119889(119909 119911) lt 119905 we have
1198652le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
int
infin
(12)119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
d120583 (119910)
120583 (119861 (119910 119889 (119909 119911)))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(37)
Finally for any 119909 isin 6119878119894 6
2
119861119894 119910 isin X and 119911 isin 6119861
119894sat-
isfying 2119889(119910 119911) le 119889(119909 119911) 2119889(119910 119911) ge 119889(119909 119911) and 119889(119909 119910) lt
(32)119889(119909 119911) by applying (2) we have
1198653le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
1
120582 (119910 119889 (119909 119911))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 119889 (119909 119911))]2
120583 (119861 (119911 (12) 119889 (119909 119911)))
120582 (119911 119889 (119909 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(38)
8 Journal of Function Spaces
Combining the estimates for 1198651 119865
2 and 119865
3 we obtain that
11986421
le 119862int6119861119894
|119891(119911)|d120583(119911) where together with the fact that119864
22le 119862int
6119861119894
|119891(119911)|d120583(119911) we have
1198642le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (39)
Now we turn to estimate for 1198641 Let 119876
119894= 119861(119888
119861119894 119903
119878119894) and
write
1198641le int
X6119878119894
[∬
119889(119909119910)lt119905
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isin119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isinX119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 11986411
+ 11986412
+ 11986413
(40)
For each fixed 119894 decompose 11986411as
11986411
le int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isin2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isinX2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 1198681+ 119868
2
(41)
For any 119909 isin X 6119878119894 119910 isin 2119878
119894with 119889(119910 119909) lt 119905 and
119911 isin 119878119894 119889(119909 119888
119861119894) minus 2119903
119878119894le 119889(119909 119910) lt 119905 and 119889(119910 119911) lt 3119903
119878119894
togetherwithMinkowski inequality and (6) we can conclude
1198681le 119862int
X6119878119894
int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
∬119889(119909119910)lt119905119889(119910119911)le119905
119910isin2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
(int
infin
119889(119909119888119861119894)minus2119903119878119894
d119905120582 (119910 119905) 119905
1+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
1
[119889 (119909 119888119861119894) minus 2119903
119878119894]
2120588
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(42)
Journal of Function Spaces 9
For 1198682 write
1198682le int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905le119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
š 11986821
+ 11986822
(43)
For 11986821 by Minkowski inequality and (6) we deduce
11986821
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
119889(119910119888119861119894)+119903119878119894
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[int
2119896+1
611987811989421198966119878119894
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(44)
Nowwe estimate 11986822 ApplyingMinkowski inequality and the
vanishing moment we have
11986822
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870(119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588+
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
+ 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909) š 1198691+ 119869
2
(45)
With a way similar to that used in the proof of 1198681 we have
1198691le 119862ℎ
1198941198711(120583) Thus we only need to estimate 119869
2 by Mink-
owski inequality and (11) it follows that
1198692le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2minus2120588
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
infin
119889(119910119888119861119894)+119903119878119894
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
10 Journal of Function Spaces
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[
[
int
2119896119903119878119894
lt119889(1199101198881198611)le2119896+1
119903119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2d120583 (119910)
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(46)
Combining the estimates for 1198691 119869
2 119868
21 and 119868
1 we obtain that
11986411
le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(47)
Next we estimate 11986412 For any 119910 isin 119861
119894 119909 isin X 6119878
119894 and
119911 isin 119878119894 we have 119889(119909 119910) ge (12)119889(119909 119888
119861119894) 119889(119910 119911) le 2119903
119878119894 and
119889(119909 119910) sim 119889(119909 119888119861119894) and together with this fact Minkowski
inequality and (6) we get
11986412
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
[
∬119889(119909119910)ge119905
119889(119910119911)le119905
119910isin119876119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(48)
It remains to estimate 11986413 Applying Minkowski inequality
and (6) we have
11986413
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
+ 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)gt2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
š 1198801+ 119880
2
(49)
Journal of Function Spaces 11
Now we estimate 1198801 For any 119910 isin X 119876
119894 119911 isin 119878
119894 and 119889(119910
119911) le 119905 le 119889(119909 119911) it is easy to see 119889(119910 119911) sim 119889(119910 119888119861119894) So we have
1198801le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
d120583 (119910)
120582 (119910 119889 (119910 119888119861119894))
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(50)
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 7
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)gt119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
d120583 (119910)
[120582 (119910 (12) 119889 (119909 119911))]
2
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 (12) 119889 (119909 119911))]2int
2119889(119910119911)gt119889(119909119911)
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))119861(1199112119896minus2
119889(119909119911))
d120583 (119910)
120582 (119910 119889 (119910 119911))
]
12
1
120582 (119911 119889 (119909 119911))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
infin
sum
119896=1
int
119861(1199112119896minus1
119889(119909119911))
d120583 (119910)
120582 (119910 2119896minus2
119889 (119909 119911))
]
12
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(35)
where we use the fact that
int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) le 119862119870119861119894 119878119894
(36)
Next we estimate 1198652 For any 119909 isin 6119878
119894 6
2
119861119894 119910 isin X
and 119911 isin 6119861119894satisfying 119889(119910 119909) lt 119905 2119889(119910 119911) le 119889(119909 119911) and
(12)119889(119909 119911) lt 119905 we have
1198652le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
int
infin
(12)119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
d120583 (119910)
120583 (119861 (119910 119889 (119909 119911)))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(37)
Finally for any 119909 isin 6119878119894 6
2
119861119894 119910 isin X and 119911 isin 6119861
119894sat-
isfying 2119889(119910 119911) le 119889(119909 119911) 2119889(119910 119911) ge 119889(119909 119911) and 119889(119909 119910) lt
(32)119889(119909 119911) by applying (2) we have
1198653le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
[
∬119889(119910119911)le119905119889(119909119910)ge119905
2119889(119910119911)le119889(119909119911)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[int
2119889(119910119911)le119889(119909119911)
1
[120582 (119910 119889 (119909 119911))]
2
1
120582 (119910 119889 (119909 119911))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
[
1
[120582 (119911 119889 (119909 119911))]2
120583 (119861 (119911 (12) 119889 (119909 119911)))
120582 (119911 119889 (119909 119911))
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816int
611987811989462119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862int
6119861119894
1003816100381610038161003816119891 (119911)
1003816100381610038161003816d120583 (119911)
(38)
8 Journal of Function Spaces
Combining the estimates for 1198651 119865
2 and 119865
3 we obtain that
11986421
le 119862int6119861119894
|119891(119911)|d120583(119911) where together with the fact that119864
22le 119862int
6119861119894
|119891(119911)|d120583(119911) we have
1198642le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (39)
Now we turn to estimate for 1198641 Let 119876
119894= 119861(119888
119861119894 119903
119878119894) and
write
1198641le int
X6119878119894
[∬
119889(119909119910)lt119905
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isin119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isinX119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 11986411
+ 11986412
+ 11986413
(40)
For each fixed 119894 decompose 11986411as
11986411
le int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isin2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isinX2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 1198681+ 119868
2
(41)
For any 119909 isin X 6119878119894 119910 isin 2119878
119894with 119889(119910 119909) lt 119905 and
119911 isin 119878119894 119889(119909 119888
119861119894) minus 2119903
119878119894le 119889(119909 119910) lt 119905 and 119889(119910 119911) lt 3119903
119878119894
togetherwithMinkowski inequality and (6) we can conclude
1198681le 119862int
X6119878119894
int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
∬119889(119909119910)lt119905119889(119910119911)le119905
119910isin2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
(int
infin
119889(119909119888119861119894)minus2119903119878119894
d119905120582 (119910 119905) 119905
1+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
1
[119889 (119909 119888119861119894) minus 2119903
119878119894]
2120588
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(42)
Journal of Function Spaces 9
For 1198682 write
1198682le int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905le119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
š 11986821
+ 11986822
(43)
For 11986821 by Minkowski inequality and (6) we deduce
11986821
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
119889(119910119888119861119894)+119903119878119894
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[int
2119896+1
611987811989421198966119878119894
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(44)
Nowwe estimate 11986822 ApplyingMinkowski inequality and the
vanishing moment we have
11986822
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870(119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588+
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
+ 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909) š 1198691+ 119869
2
(45)
With a way similar to that used in the proof of 1198681 we have
1198691le 119862ℎ
1198941198711(120583) Thus we only need to estimate 119869
2 by Mink-
owski inequality and (11) it follows that
1198692le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2minus2120588
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
infin
119889(119910119888119861119894)+119903119878119894
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
10 Journal of Function Spaces
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[
[
int
2119896119903119878119894
lt119889(1199101198881198611)le2119896+1
119903119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2d120583 (119910)
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(46)
Combining the estimates for 1198691 119869
2 119868
21 and 119868
1 we obtain that
11986411
le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(47)
Next we estimate 11986412 For any 119910 isin 119861
119894 119909 isin X 6119878
119894 and
119911 isin 119878119894 we have 119889(119909 119910) ge (12)119889(119909 119888
119861119894) 119889(119910 119911) le 2119903
119878119894 and
119889(119909 119910) sim 119889(119909 119888119861119894) and together with this fact Minkowski
inequality and (6) we get
11986412
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
[
∬119889(119909119910)ge119905
119889(119910119911)le119905
119910isin119876119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(48)
It remains to estimate 11986413 Applying Minkowski inequality
and (6) we have
11986413
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
+ 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)gt2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
š 1198801+ 119880
2
(49)
Journal of Function Spaces 11
Now we estimate 1198801 For any 119910 isin X 119876
119894 119911 isin 119878
119894 and 119889(119910
119911) le 119905 le 119889(119909 119911) it is easy to see 119889(119910 119911) sim 119889(119910 119888119861119894) So we have
1198801le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
d120583 (119910)
120582 (119910 119889 (119910 119888119861119894))
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(50)
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
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Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Journal of Function Spaces
Combining the estimates for 1198651 119865
2 and 119865
3 we obtain that
11986421
le 119862int6119861119894
|119891(119911)|d120583(119911) where together with the fact that119864
22le 119862int
6119861119894
|119891(119911)|d120583(119911) we have
1198642le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (39)
Now we turn to estimate for 1198641 Let 119876
119894= 119861(119888
119861119894 119903
119878119894) and
write
1198641le int
X6119878119894
[∬
119889(119909119910)lt119905
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isin119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)ge119905
119910isinX119876119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 11986411
+ 11986412
+ 11986413
(40)
For each fixed 119894 decompose 11986411as
11986411
le int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isin2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905
119910isinX2119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909) š 1198681+ 119868
2
(41)
For any 119909 isin X 6119878119894 119910 isin 2119878
119894with 119889(119910 119909) lt 119905 and
119911 isin 119878119894 119889(119909 119888
119861119894) minus 2119903
119878119894le 119889(119909 119910) lt 119905 and 119889(119910 119911) lt 3119903
119878119894
togetherwithMinkowski inequality and (6) we can conclude
1198681le 119862int
X6119878119894
int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
∬119889(119909119910)lt119905119889(119910119911)le119905
119910isin2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
(int
infin
119889(119909119888119861119894)minus2119903119878119894
d119905120582 (119910 119905) 119905
1+2120588
) d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int
119889(119910119911)le3119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
1
[119889 (119909 119888119861119894) minus 2119903
119878119894]
2120588
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[int
119889(119910119911)le3119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120583 (119861 (119910 119889 (119909 119888119861119894)))
d120583 (119910)]
12
d120583 (119909) d120583 (119911)
le 119862int
6119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(42)
Journal of Function Spaces 9
For 1198682 write
1198682le int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905le119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
š 11986821
+ 11986822
(43)
For 11986821 by Minkowski inequality and (6) we deduce
11986821
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
119889(119910119888119861119894)+119903119878119894
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[int
2119896+1
611987811989421198966119878119894
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(44)
Nowwe estimate 11986822 ApplyingMinkowski inequality and the
vanishing moment we have
11986822
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870(119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588+
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
+ 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909) š 1198691+ 119869
2
(45)
With a way similar to that used in the proof of 1198681 we have
1198691le 119862ℎ
1198941198711(120583) Thus we only need to estimate 119869
2 by Mink-
owski inequality and (11) it follows that
1198692le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2minus2120588
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
infin
119889(119910119888119861119894)+119903119878119894
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
10 Journal of Function Spaces
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[
[
int
2119896119903119878119894
lt119889(1199101198881198611)le2119896+1
119903119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2d120583 (119910)
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(46)
Combining the estimates for 1198691 119869
2 119868
21 and 119868
1 we obtain that
11986411
le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(47)
Next we estimate 11986412 For any 119910 isin 119861
119894 119909 isin X 6119878
119894 and
119911 isin 119878119894 we have 119889(119909 119910) ge (12)119889(119909 119888
119861119894) 119889(119910 119911) le 2119903
119878119894 and
119889(119909 119910) sim 119889(119909 119888119861119894) and together with this fact Minkowski
inequality and (6) we get
11986412
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
[
∬119889(119909119910)ge119905
119889(119910119911)le119905
119910isin119876119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(48)
It remains to estimate 11986413 Applying Minkowski inequality
and (6) we have
11986413
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
+ 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)gt2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
š 1198801+ 119880
2
(49)
Journal of Function Spaces 11
Now we estimate 1198801 For any 119910 isin X 119876
119894 119911 isin 119878
119894 and 119889(119910
119911) le 119905 le 119889(119909 119911) it is easy to see 119889(119910 119911) sim 119889(119910 119888119861119894) So we have
1198801le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
d120583 (119910)
120582 (119910 119889 (119910 119888119861119894))
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(50)
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 9
For 1198682 write
1198682le int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905le119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
+ int
X6119878119894
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
(
119905
119905 + 119889 (119909 119910)
)
120581 1003816100381610038161003816100381610038161003816100381610038161003816
1
119905120588
int
119889(119910119911)le119905
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
ℎ119894(119911) d120583 (119911)
1003816100381610038161003816100381610038161003816100381610038161003816
2 d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119909)
š 11986821
+ 11986822
(43)
For 11986821 by Minkowski inequality and (6) we deduce
11986821
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
119889(119910119888119861119894)+119903119878119894
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119910isinX2119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[int
2119896+1
611987811989421198966119878119894
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
d120583 (119910)]
12
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(44)
Nowwe estimate 11986822 ApplyingMinkowski inequality and the
vanishing moment we have
11986822
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870(119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588+
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
le 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119911)]
1minus120588
minus
119870 (119910 119911)
[119889 (119910 119888119861119894)]
1minus120588
)ℎ119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909)
+ 119862int
X6119878119894
[
[
[
∬119889(119909119910)lt119905119910isinX2119878119894
119905gt119889(119910119888119861119894)+119903119878119894
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
int
119889(119910119911)le119905
(
119870(119910 119911)
[119889 (119910 119888119861119894)]
1minus120588minus
119870 (119910 119888119861119894)
[119889 (119910 119888119861119894)]
1minus120588)ℎ
119894(119911) d120583 (119911)
100381610038161003816100381610038161003816100381610038161003816100381610038161003816
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119909) š 1198691+ 119869
2
(45)
With a way similar to that used in the proof of 1198681 we have
1198691le 119862ℎ
1198941198711(120583) Thus we only need to estimate 119869
2 by Mink-
owski inequality and (11) it follows that
1198692le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2minus2120588
1
120582 (119910 119889 (119910 119888119861119894) + 119903
119878119894)
(int
infin
119889(119910119888119861119894)+119903119878119894
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
10 Journal of Function Spaces
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[
[
int
2119896119903119878119894
lt119889(1199101198881198611)le2119896+1
119903119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2d120583 (119910)
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(46)
Combining the estimates for 1198691 119869
2 119868
21 and 119868
1 we obtain that
11986411
le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(47)
Next we estimate 11986412 For any 119910 isin 119861
119894 119909 isin X 6119878
119894 and
119911 isin 119878119894 we have 119889(119909 119910) ge (12)119889(119909 119888
119861119894) 119889(119910 119911) le 2119903
119878119894 and
119889(119909 119910) sim 119889(119909 119888119861119894) and together with this fact Minkowski
inequality and (6) we get
11986412
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
[
∬119889(119909119910)ge119905
119889(119910119911)le119905
119910isin119876119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(48)
It remains to estimate 11986413 Applying Minkowski inequality
and (6) we have
11986413
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
+ 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)gt2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
š 1198801+ 119880
2
(49)
Journal of Function Spaces 11
Now we estimate 1198801 For any 119910 isin X 119876
119894 119911 isin 119878
119894 and 119889(119910
119911) le 119905 le 119889(119909 119911) it is easy to see 119889(119910 119911) sim 119889(119910 119888119861119894) So we have
1198801le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
d120583 (119910)
120582 (119910 119889 (119910 119888119861119894))
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(50)
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
Volume 2014
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Journal of Function Spaces
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int
X2119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
sdot int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
infin
sum
119896=1
[
[
int
2119896119903119878119894
lt119889(1199101198881198611)le2119896+1
119903119878119894
10038161003816100381610038161003816119870 (119910 119911) minus 119870 (119910 119888
119861119894)
10038161003816100381610038161003816
2 1
[119889 (119910 119888119861119894)]
2d120583 (119910)
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(46)
Combining the estimates for 1198691 119869
2 119868
21 and 119868
1 we obtain that
11986411
le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(47)
Next we estimate 11986412 For any 119910 isin 119861
119894 119909 isin X 6119878
119894 and
119911 isin 119878119894 we have 119889(119909 119910) ge (12)119889(119909 119888
119861119894) 119889(119910 119911) le 2119903
119878119894 and
119889(119909 119910) sim 119889(119909 119888119861119894) and together with this fact Minkowski
inequality and (6) we get
11986412
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
[
∬119889(119909119910)ge119905
119889(119910119911)le119905
119910isin119876119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
1+2120588
]
]
]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816[int
119889(119910119911)le2119903119878119894
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119888119861119894))
d120583 (119910)]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(48)
It remains to estimate 11986413 Applying Minkowski inequality
and (6) we have
11986413
le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
+ 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119909)
1003816100381610038161003816
[
[
∬119889(119910119911)le119905le119889(119909119910)119910isinX119876119894
119889(119909119888119861119894)gt2119889(119910119888119861119894
)
(
119905
119905 + 119889 (119909 119910)
)
120581
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
d120583 (119910) d119905120582 (119910 119905) 119905
]
]
12
d120583 (119911) d120583 (119909)
š 1198801+ 119880
2
(49)
Journal of Function Spaces 11
Now we estimate 1198801 For any 119910 isin X 119876
119894 119911 isin 119878
119894 and 119889(119910
119911) le 119905 le 119889(119909 119911) it is easy to see 119889(119910 119911) sim 119889(119910 119888119861119894) So we have
1198801le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
d120583 (119910)
120582 (119910 119889 (119910 119888119861119894))
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(50)
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Function Spaces 11
Now we estimate 1198801 For any 119910 isin X 119876
119894 119911 isin 119878
119894 and 119889(119910
119911) le 119905 le 119889(119909 119911) it is easy to see 119889(119910 119911) sim 119889(119910 119888119861119894) So we have
1198801le 119862int
X6119878119894
int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
[119889 (119910 119911)]
2120588
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
(int
119889(119909119910)
119889(119910119911)
d1199051199051+2120588
) d120583 (119910)]
]
12
d120583 (119911) d120583 (119909)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
1
[120582 (119910 119889 (119910 119911))]
2
1
120582 (119910 119889 (119909 119910))
d120583 (119910)]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
[
[
int119910isinX119876119894
119889(119909119888119861119894)le2119889(119910119888119861119894
)
d120583 (119910)
120582 (119910 119889 (119910 119888119861119894))
]
]
12
d120583 (119909) d120583 (119911)
le 119862int
119878119894
1003816100381610038161003816ℎ119894(119911)
1003816100381610038161003816int
X6119878119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909) d120583 (119911) le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(50)
On the other hand by a method similar to that used inthe proof of 119880
1 we have
1198802le 119862
1003817100381710038171003817ℎ119894
10038171003817100381710038171198711(120583)
(51)
Combining the estimates 1198801 119880
2 119864
11 119864
12 and the fact that
ℎ1198941198711(120583)
le 119862int6119861119894
|119891(119909)|d120583(119909) we conclude that
1198641le 119862int
6119861119894
1003816100381610038161003816119891 (119909)
1003816100381610038161003816d120583 (119909) (52)
which together with 1198642 implies (30) and the proof of
Theorem 10 is finished
Proof ofTheorem 11 Without loss of generality we assume 120577 =
2 By a standard argument it suffices to show that for any(infin 1)-atomic block 119887
1003817100381710038171003817M
lowast120588
120581(119887)
10038171003817100381710038171198711(120583)
le 119862 |119887|1198671infin
atb (120583) (53)
Assume that supp 119887 sub 119877 and 119887 = sum2
119894=1120592119894119886119894 where for
119894 isin 1 2 119886119894is a function supported in 119861
119894sub 119877 such that
119886119894119871infin(120583)
le [120583(4119861119894)]
minus1
119870minus1
119861119894 119877and |120592
1| + |120592
2| sim |119887|
1198671infin
atb (120583) Write
int
X
Mlowast120588
120581(119887) (119909) d120583 (119909)
= int
2119877
Mlowast120588
120581(119887) (119909) d120583 (119909)
+ int
X2119877
Mlowast120588
120581(119887) (119909) d120583 (119909) š 119881
1+ 119881
2
(54)
For 1198811 we see that
1198811le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
2119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
+
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816int
21198772119861119894
Mlowast120588
120581(119886
119894) (119909) d120583 (119909)
š 11988111
+ 11988112
(55)
Applying the Holder inequality 1198712
(120583)-boundedness ofMlowast120588
120581
and the fact that 119886119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877for 119894 isin 1 2
we have
11988111
le
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816(int
2119861119894
1003816100381610038161003816M
lowast120588
120581(119886
119894) (119909)
1003816100381610038161003816
2 d120583 (119909))
12
120583 (2119861119894)
12
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198712(120583)
120583 (2119861119894)
12
le 119862 |119887|1198671infin
atb (120583)
(56)
Now we estimate 11988112 with a method similar to that used in
the proof of 1198651and 119886
119894119871infin(120583)
le 119862[120583(4119861119894)]
minus1
119870minus1
119861119894 119877 and we see
that
11988112
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816
1003817100381710038171003817119886119894
10038171003817100381710038171198711(120583)
int
21198772119861119894
1
120582 (119888119861119894 119889 (119909 119888
119861119894))
d120583 (119909)
le 119862
2
sum
119894=1
1003816100381610038161003816120592119894
1003816100381610038161003816119870
119861119894 119877
1003817100381710038171003817119886119894
1003817100381710038171003817119871infin(120583)
120583 (119861119894) le 119862 |119887|
1198671infin
atb (120583)
(57)
Therefore 1198811le 119862|119887|
1198671infin
atb (120583)
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Journal of Function Spaces
On the other hand based on the proof of 1198641and
Definition 8 it is easy to obtain that
1198812le 119862 119887
1198711(120583)
le 119862 |119887|1198671infin
atb (120583) (58)
Combining the estimates for 1198811and 119881
2 (53) holds Thus
Theorem 11 is completed
Competing Interests
The authors declare that there are no competing interestsregarding the publication of this paper
Authorsrsquo Contributions
All authors contributed equally to the writing of this paperAll authors read and approved the final paper
Acknowledgments
This paper is supported by National Natural Foundation ofChina (Grant no 11561062)
References
[1] E M Stein ldquoOn the functions of Littlewood-Paley Lusin andMarcinkiewiczrdquo Transactions of the American MathematicalSociety vol 88 no 2 pp 430ndash466 1958
[2] C Fefferman ldquoInequalities for strongly singular convolutionoperatorsrdquo Acta Mathematica vol 124 pp 9ndash36 1970
[3] M Sakamoto and K Yabuta ldquoBoundedness of marcinkiewiczfunctionsrdquo StudiaMathematica vol 135 no 2 pp 103ndash142 1999
[4] Y Ding S Lu and K Yabuta ldquoA problem on rough parametricMarcinkiewicz functionsrdquo Journal of the Australian Mathemat-ical Society vol 72 no 1 pp 13ndash21 2002
[5] Y Ding and Q Xue ldquoEndpoint estimates for commutators ofa class of Littlewood-Paley operatorsrdquo Hokkaido MathematicalJournal vol 36 no 2 pp 245ndash282 2007
[6] L Wang and S Tao ldquoBoundedness of Littlewood-Paley oper-ators and their commutators on Herz-Morrey spaces withvariable exponentrdquo Journal of Inequalities and Applications vol2014 article 227 2014
[7] X Tolsa ldquoBMO H1 and Calderon-Zygmund operator for non-doubling measuresrdquoMathematische Annalen vol 319 no 1 pp89ndash149 2001
[8] H Lin andYMeng ldquoBoundedness of parametrized Littlewood-Paley operators with nondoubling measuresrdquo Journal ofInequalities and Applications vol 2008 Article ID 141379 25pages 2008
[9] X Tolsa ldquoThe spaces H1 for non-doubling measures in termsof a grand maximal operatorrdquo Transactions of the AmericanMathematical Society vol 355 no 1 pp 315ndash348 2003
[10] G Hu H Lin and D Yang ldquoMarcinkiewicz integrals with non-doubling measuresrdquo Integral Equations and Operator Theoryvol 58 no 2 pp 205ndash238 2007
[11] Q Xue and J Zhang ldquoEndpoint estimates for a class ofLittlewood-Paley operators with nondoubling measuresrdquo Jour-nal of Inequalities and Applications vol 2009 Article ID 17523028 pages 2009
[12] X Tolsa ldquoLittlewood-Paley theory and the T(1) theorem withnon-doublingmeasuresrdquoAdvances inMathematics vol 164 no1 pp 57ndash116 2001
[13] X Tolsa ldquoPainleversquos problem and the semiadditivity of analyticcapacityrdquo Acta Mathematica vol 190 no 1 pp 105ndash149 2003
[14] X Tolsa ldquoThe semiadditivity of continuous analytic capacityand the inner boundary conjecturerdquo American Journal ofMathematics vol 126 no 3 pp 523ndash567 2004
[15] R R Coifman and G Weiss Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes vol 242 ofLecture Notes in Mthematics Springer Berlin Germany 1971
[16] R R Coifman and G Weiss ldquoExtensions of Hardy spaces andtheir use in analysisrdquo Bulletin of the American MathematicalSociety vol 83 no 4 pp 569ndash645 1977
[17] T Hytonen ldquoA framework for non-homogeneous analysis onmetric spaces and the RBMO space of Tolsardquo PublicacionsMatematiques vol 54 no 2 pp 485ndash504 2010
[18] THtyonen D Yang andD Yang ldquoTheHardy spaceH1 on non-homogeneous metric spacesrdquo Mathematical Proceedings of theCambridge Philosophical Society vol 153 no 1 pp 9ndash31 2012
[19] X Fu D Yang and W Yuan ldquoGeneralized fractional integralsand their commutators over non-homogeneousmetricmeasurespacesrdquo Taiwanese Journal of Mathematics vol 18 no 2 pp509ndash557 2014
[20] G Lu and S Tao ldquoBoundedness of commutators ofMarcinkiewicz integrals on nonhomogeneous metric measurespacesrdquo Journal of Function Spaces vol 2015 Article ID 54816512 pages 2015
[21] T A Bui and X T Duong ldquoHardy spaces regularized BMOspaces and the boundedness of Caldern-Zygmund operators onnon-homogeneous spacesrdquo Journal of Geometric Analysis vol23 no 2 pp 895ndash932 2013
[22] H Lin andD Yang ldquoEquivalent boundedness ofMarcinkiewiczintegrals on non-homogeneous metric measure spacesrdquo ScienceChina Mathematics vol 57 no 1 pp 123ndash144 2014
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of