essential norms of volterra type operators between zygmund

Research ArticleEssential Norms of Volterra Type Operators betweenZygmund Type Spaces

Shanli Ye1 and Caishu Lin2

1School of Science Zhejiang University of Science and Technology Hangzhou 310023 China2Department of Mathematics Fujian Normal University Fuzhou 350007 China

Correspondence should be addressed to Shanli Ye ye_shanlialiyuncom

Received 5 January 2017 Revised 4 March 2017 Accepted 7 March 2017 Published 11 April 2017

Academic Editor Ruhan Zhao

Copyright copy 2017 Shanli Ye and Caishu Lin This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We investigate the boundedness of some Volterra type operators between 119885119910119892119898119906119899119889 type spacesThen we give the essential normsof such operators in terms of 119892 120593 their derivatives and the nth power 120593119899 of 120593

1 Introduction

Let 119863 = 119911 |119911| lt 1 be the open unit disk in the complexplane C and let 120597119863 = 119911 |119911| = 1 be its boundary and119867(119863)denote the set of all analytic functions on119863

For every 0 lt 120572 lt infin we denote by B120572 the Bloch typespace of all functions 119891 isin 119867(119863) satisfying

119887120572 (119891) = sup119911isin119863

(1 minus |119911|2)120572 100381610038161003816100381610038161198911015840 (119911)10038161003816100381610038161003816 lt infin (1)

endowed with the norm 119891B120572 = |119891(0)| + 119887120572(119891) The littleBloch type space B1205720 consists of all 119891 isin B120572 satisfyinglim|119911|rarr1minus(1minus|119911|2)120572|1198911015840(119911)| = 0 andB1205720 is obviously the closedsubspace ofB120572 When 120572 = 1 we get the classical Bloch spaceB1 = B and little Bloch space B10 = B0 It is well knownthat for 0 lt 120572 lt 1 B120572 is a subspace of 119867infin the Banachspace of bounded analytic functions on 119863 Some sources forresults and references about the Bloch type functions are thepapers of Yoneda [1] Stevic [2 3] and the first author [4ndash7]

For 0 lt 120572 lt infin we denote byZ120572 the Zygmund type spaceof those functions 119891 isin 119867(119863) satisfying

sup119911isin119863

(1 minus |119911|2)120572 1003816100381610038161003816100381611989110158401015840 (119911)10038161003816100381610038161003816 lt infin (2)

and the little Zygmund type spaceZ1205720 consists of all 119891 isin Z120572

satisfying lim|119911|rarr1minus(1minus|119911|2)120572|11989110158401015840(119911)| = 0 andZ1205720 is obviouslythe closed subspace ofZ120572 It can easily be proved thatZ120572 isa Banach space under the norm

10038171003817100381710038171198911003817100381710038171003817Z120572 = 1003816100381610038161003816119891 (0)1003816100381610038161003816 + 100381610038161003816100381610038161198911015840 (0)10038161003816100381610038161003816 + sup119911isin119863

(1 minus |119911|2)120572 1003816100381610038161003816100381611989110158401015840 (119911)10038161003816100381610038161003816 (3)

and thatZ1205720 is a closed subspace ofZ120572 When 120572 = 1 we getthe classical Zygmund spaceZ1 = Z and the little ZygmundspaceZ10 = Z0 It is clear that 119891 isin Z if and only if 1198911015840 isin B1

We consider the weighted Banach spaces of analyticfunctions

119867infinV = 119891 isin 119867 (119863) 119865V = sup119911isin119863

V (119911) 1003816100381610038161003816119891 (119911)1003816100381610038161003816 lt infin (4)

endowed with norm sdot V where the weight V 119863 rarr 119877+is a continuous strictly positive and bounded function Theweight V is called radial if V(119911) = V(|119911|) for all 119911 isin 119863 For aweight V the associated weight V is defined by

V (119911) = (sup 1003816100381610038161003816119891 (119911)1003816100381610038161003816 119891 isin 119867infinV 10038171003817100381710038171198911003817100381710038171003817V le 1)minus1 119911 isin 119863 (5)

HindawiJournal of Function SpacesVolume 2017 Article ID 1409642 14 pageshttpsdoiorg10115520171409642

2 Journal of Function Spaces

We notice the standard weights V120572(119911) = (1minus |119911|2)120572 where 0 lt120572 lt infin and it is well known that V120572 = V120572 We also considerthe logarithmic weight

Vlog = (log 21 minus |119911|2)minus1 119911 isin 119863 (6)

It is straightforward to show that Vlog = VlogFor an analytic self-map 120593 of119863 and a function 119906 isin 119867(119863)

we define the weighted composition operator as 119906119862120593119891 =119906 sdot (119891 ∘ 120593) for 119891 isin 119867(119863) Weighted composition operatorshave been extensively studied recently It is interesting toprovide a function theoretic characterization when 120593 and119906 induce a bounded or compact composition operator onvarious function spaces Some results on the boundednessand compactness of concrete operators between some spacesof analytic functions one of which is of Zygmund type can befound for example in [8ndash19]

Suppose that 119892 119863 rarr C is an analytic map Let 119879119892 and 119868119892denote the Volterra type operators with the analytic symbol119892 on119863 respectively

119879119892119891 (119911) = int1199110119891 (120585) 1198921015840 (120585) 119889120585 119911 isin 119863

119868119892119891 (119911) = int11991101198911015840 (120585) 119892 (120585) 119889120585 119911 isin 119863

(7)

If 119892(119911) = 119911 then 119879119892 is an integral operator While119892(119911) = ln(1(1minus119911)) then119879119892 is Cesaro operator Pommerenkeintroduced the 119881119900119897119905119890119903119903119886 type operator 119879119892 and characterizedthe boundedness of 119879119892 between 1198672 spaces in [20] Morerecently boundedness and compactness of 119881119900119897119905119890119903119903119886 typeoperators between several spaces of analytic functions havebeen studied by many authors one may see [21 22]

In this paper we consider the following integral typeoperators which were introduced by Li and Stevic (see eg[10 23]) they can be defined by

(119862120593119879119892119891) (119911) = int120593(119911)0

119891 (120585) 1198921015840 (120585) 119889120585(119862120593119868119892119891) (119911) = int120593(119911)

01198911015840 (120585) 119892 (120585) 119889120585

(119879119892119862120593119891) (119911) = int1199110119891 (120593 (120585)) 1198921015840 (120585) 119889120585

(119868119892119862120593119891) (119911) = int11991101198911015840 (120593 (120585)) 119892 (120585) 119889120585

(8)

We will characterize the boundedness of those integral typeoperators between Zygmund type spaces and also estimatetheir essential norms The boundedness and compactness ofthese operators on the logarithmic Bloch space have beencharacterized in [22]

Recall that essential norm 119879119890119883rarr119884 of a bounded linearoperator 119879 119883 rarr 119884 is defined as the distance from 119879

to K(119883 119884) the space of compact operators from 119883 to 119884namely

119879119890119883rarr119884= inf 119879 + 119870119883rarr119884 119870 119883 rarr 119884 is compact (9)

It provides a measure of noncompactness of 119879 Clearly 119879 iscompact if and only if 119879119890119883rarr119884 = 0

Throughout this paper constants are denoted by 119862 theyare positive and may differ from one occurrence to the otherThe notation 119886 ≍ 119887 means that there are positive constants1198621 1198622 such that 1198621119886 le 119887 le 11986221198862 Boundedness

In order to prove themain results of this paperWe need someauxiliary results

Lemma 1 (see [8 13]) For 0 lt 120572 lt 2 and let 119891119899 be a boundedsequence in Z120572 which converges to 0 uniformly on compactsubsets of119863 Then lim119899rarrinfinsup119911isin119863|119891119899(119911)| = 0Lemma 2 (see [8 13]) For every 119891 isin Z120572 where 120572 gt 0 onehas

(i) |1198911015840(119911)| le (2(1 minus 120572))119891Z120572 and |119891(119911)| le (2(1 minus120572))119891Z120572 for every 0 lt 120572 lt 1(ii) |1198911015840(119911)| le 2 log(2(1 minus |119911|))119891Z120572 and |119891(119911)| le 119891Z120572

for 120572 = 1(iii) |1198911015840(119911)| le (2(1 minus 120572))(119891Z120572(1 minus |119911|)120572minus1) for every120572 gt 1(iv) |119891(119911)| le (2(120572 minus 1)(2 minus 120572))119891Z120572 for every 1 lt 120572 lt 2(v) |119891(119911)| le 2 log(2(1 minus |119911|))119891Z120572 for every 120572 = 2(vi) |119891(119911)| le (2(120572 minus 1)(120572 minus 2))(119891Z120572(1 minus |119911|)120572minus2) for

every 120572 gt 2Lemma 3 (see [8]) Let 0 lt 120572 lt infin and V a radial non-increasing weight tending to 0 at boundary of 119863 and let theweighted composition operator 119906119862120593 Z120572 rarr 119867infinV be bounded

(i) If 0 lt 120572 lt 2 then 119906119862120593 is a compact operator(ii)1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890Z2rarr119867infinV ≍ lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171199061205931198991003817100381710038171003817V

≍ lim sup|120593(119911)|rarr1

V (119911) |119906 (119911)| log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(10)

(iii) If 120572 gt 2 then1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890Z120572rarr119867infinV ≍ lim sup

119899rarrinfin(119899 + 1)120572minus2 10038171003817100381710038171199061205931198991003817100381710038171003817V

≍ lim sup|120593(119911)|rarr1

V (119911) |119906 (119911)|(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2

(11)

The following lemma is due to [24 25]

Journal of Function Spaces 3

Lemma 4 Let V and 119908 be radial nonincreasing weightstending to zero at the boundary of119863 Then

(i) Theweighted composition operator 119906119862120593 maps119867infinV into119867infin119908 if and only if

sup119899ge0

10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V ≍ sup119911isin119863

119908 (119911)V (120593 (119911)) |119906 (119911)| lt infin (12)

with norm comparable to the above supermum

(ii)

1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890119867infinV rarr119867infin119908 = lim sup119899rarrinfin

10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V= lim sup|120593(119911)|rarr1

119908 (119911)V (120593 (119911)) |119906 (119911)|

(13)

Lemma 5 (see [26]) For every 0 lt 120572 lt infin one has

lim119899rarrinfin

(119899 + 1)120572 10038171003817100381710038171199111198991003817100381710038171003817V120572 = (2120572119890 )120572 (14)

lim119899rarrinfin

(log 119899) 10038171003817100381710038171199111198991003817100381710038171003817Vlog = 1 (15)

Theorem6 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded

operator if and only if 1198921015840 isin 119867infinV120573 andsup119899ge0

(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (16)

(ii) If 120572 = 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if

sup119899ge0

(119899 + 1) 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin(17)

sup119899ge0

(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (18)

(iii) If 120572 gt 1 then 119868119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if

sup119899ge0

(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (19)

sup119899ge0

(119899 + 1)120572minus1 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1

100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (20)

Proof Suppose that 119868119892119862120593 is bounded fromZ120572 toZ120573 Usingthe test functions 119891(119911) = 119911 and 119891(119911) = 1199112 we have

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119868119892119862120593119911)10158401015840100381610038161003816100381610038161003816 = (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (21)

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1198681198921198621205931199112)10158401015840100381610038161003816100381610038161003816= (1 minus |119911|2)120573 1003816100381610038161003816100381621205931015840 (119911) 119892 (119911) + 2120593 (119911) 1198921015840 (119911)10038161003816100381610038161003816 lt infin (22)

Since 120593 is a self-map we get that 1198921015840 isin 119867infinV120573 1205931015840119892 isin 119867infinV120573 For every 0 lt 120572 lt infin and given nonzero 119886 isin 119863 we take

the test functions

119891119886 (119911) = 11198862 [[

(1 minus |119886|2)2(1 minus 119886119911)120572 minus 1 minus |119886|2(1 minus 119886119911)120572minus1]] (23)

ℎ119886 (119911) = 1119886 int1199110

1 minus |119886|2(1 minus 119886119908)120572 119889119908 (24)

119892119886 (119911) = 119891119886 (119911) minus ℎ119886 (119911) (25)

for every 119911 isin 119863 One can show that 119891119886 ℎ119886 and 119892119886 are inZ1205720 sup12lt119886lt1119891119886Z120572 lt infin and sup12lt119886lt1ℎ119886Z120572 lt infin Since1198921015840119886(119886) = 0 11989210158401015840119886 (119886) = 120572(1 minus |119886|2)120572 it follows that for all 119911 isin 119863with |120593(119911)| gt 12 we have

+infin gt 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 ge 10038171003817100381710038171003817119868119892119862120593 (119892120593(119911))10038171003817100381710038171003817Z120573ge (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989210158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816

minus (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840120593(119911) (120593 (119911))10038161003816100381610038161003816= (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816100381610038161003816120572

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(26)


Then

sup119911isin119863

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

le sup|120593(119911)|le12

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

+ sup|120593(119911)|gt12

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

le (43)120572 10038171003817100381710038171003817120593101584011989210038171003817100381710038171003817V120573 + 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 lt infin

(27)

Now we use (14) and Lemma 4 to conclude that

sup119899ge0

(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (28)

which shows that (16) is necessary for all caseConversely suppose that 1198921015840 isin 119867infinV120573 and (16) holds As-

sume that 119891 isin Z120572 From Lemma 2 it follows that

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119868119892119862120593119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573sdot 100381610038161003816100381610038161205931015840 (119911) 119892 (119911) 11989110158401015840 (120593 (119911)) + 1198921015840 (119911) 1198911015840 (120593 (119911))10038161003816100381610038161003816le (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911) 11989110158401015840 (120593 (119911))10038161003816100381610038161003816 + (1 minus |119911|2)120573

sdot 100381610038161003816100381610038161198921015840 (119911) 1198911015840 (120593 (119911))10038161003816100381610038161003816 le (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + 119862 (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572 10038161003816100381610038161003816119868119892119862120593 (119891) (0)10038161003816100381610038161003816 = 0

100381610038161003816100381610038161003816(119868119892119862120593119891)1015840 (0)100381610038161003816100381610038161003816 = 100381610038161003816100381610038161198911015840 (120593 (0)) 119892 (0)10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572sdot 1003816100381610038161003816119892 (120593 (0))1003816100381610038161003816

(29)

which implies that 119868119892119862120593 is boundedThis completes the proofof (i)

Next we will prove (ii)The necessity in condition (17) hasbeen proved above Fixing 119886 isin 119863 with |119886| gt 12 we take thefunction

119896119886 (119911) = 119901 (119886119911)119886 (log 11 minus |119886|)minus1 (30)

for 119911 isin 119863 where

119901 (119911) = (119911 minus 1) ((1 + log 11 minus 119911)2 + 1) (31)

Then we have sup12lt|119886|lt1119896119886Z120572 le 119862 by [11] Let 119886 = 120593(119911) Itfollows that10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573

ge (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 100381610038161003816100381610038161198961015840120593(119911) (120593 (119911))10038161003816100381610038161003816minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 1003816100381610038161003816100381611989610158401015840120593(119911) (120593 (119911))10038161003816100381610038161003816

= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(32)

Since (17) holds and 119868119892119862120593 is bounded we obtain that

sup|120593(119911)|gt12

(1 minus |119911|2)120573 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ sup|120593(119911)|gt12

10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573 lt infin(33)

Noting 1198921015840 isin 119867infinV120573 and together with (15) and Lemma 4 weconclude that (18) holds

The converse implication can be shown as in the proof of(i)

Finally we will prove (iii) We have proved that (19) holdsabove To prove (20) we take function 119891120593(119911) defined in (23)for every 119911 isin 119863 with |120593(119911)| gt 12 and obtain that10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573


= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 110038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816

1003816100381610038161003816100381610038161003816100381610038161003816100381610038162120593 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(34)

Since 119868119892119862120593 is bounded and (19) holds we obtain that

sup|120593(119911)|gt12


100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le 2 sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ 2 sup|120593(119911)|gt12

10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(35)

therefore we deduce that (20) holds by (14) and Lemma 4



Theorem 7 Let 120593 be an analytic self-map of119863 and 119892 isin 119867(119863)(i) If 0 lt 120572 lt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded

operator if and only if ((119892∘120593)(12059310158401015840)+(1198921015840∘120593)(1205931015840)2) isin 119867infinV120573and

sup119899ge0

(119899 + 1)120572 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(36)


sup119899ge0

(119899 + 1) 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

sdot 100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infinsup119899ge0

(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin

(37)

(iii) If 120572 gt 1 then 119862120593119868119892 Z120572 rarr Z120573 is a bounded operatorif and only if (36) holds and

sup119899ge0

(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(38)

The proof is similar to that of Theorem 6 and the detailsare omitted


operator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573

(ii) If 120572 = 1 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and

sup119899ge0

(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)120573

sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (39)

(iii) If 1 lt 120572 lt 2 then 119862120593119879119892 Z120572 rarr Z120573 is a boundedoperator if and only if ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin119867infinV120573 and

sup119899ge0

(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (40)

(iv) If 120572 = 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if

sup119899ge0

(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(41)

sup119899ge0

(log 119899) 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log( 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162))

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(42)

(v) If 120572 gt 2 then119862120593119879119892 Z120572 rarr Z120573 is a bounded operatorif and only if (40) holds and

sup119899ge0

(119899 + 1)120572minus2 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2

+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(43)


Proof Suppose that 119862120593119879119892 is bounded fromZ120572 toZ120573 space

(i) Case 0 lt 120572 lt 1 Using functions 119891 = 1 isin Z120572 and 119891 = 119911 isinZ120572 we obtain

sup119911isin119863

(1 minus |119911|2)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infinsup119911isin119863

(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)

Then we obtain that ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and(1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 are necessary for all caseFor the converse implication suppose that ((1198921015840 ∘120593)(12059310158401015840)+(11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and (1198921015840 ∘ 120593)(1205931015840)2 isin 119867infinV120573 For 119891 isin Z120572 it

follows from Lemma 2 that

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119891)10158401015840 (119911)100381610038161003816100381610038161003816 = (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2sdot 1198921015840 (120593 (119911)) 1198911015840 (120593 (119911))+ (11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))sdot 1198911015840 (120593 (119911))100381610038161003816100381610038161003816 + (1 minus |119911|2)120573sdot 100381610038161003816100381610038161003816(11989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911))sdot 119891 (120593 (119911))100381610038161003816100381610038161003816 le (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + (1 minus |119911|2)120573 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572

10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)

0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816

le max|120577|le|120593(0)|

1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|

100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 le 21 minus 120572 10038171003817100381710038171198911003817100381710038171003817Z120572sdot max|120577|le|120593(0)|

100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572

sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816

(45)

Then 119862120593119879119892 is bounded This completes the proof of (i)

(ii) Case 120572 = 1 We consider the test function 119896120593(119911)(119911) definedin (30) for every 119911 isin 119863 with |120593(119911)| gt 12 It follows that

10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 ge log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 minus (1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816

(46)

Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 andsup|120593(119911)|gt12119862120593119879119892119896120593(119911)Z120573 le 119862 we get

sup|120593(119911)|gt12

log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12

(1 minus |119911|2)120573sdot 1003816100381610038161003816100381612059310158401015840 (119911) 1198921015840 (120593 (119911)) + 11989210158401015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816sdot 10038161003816100381610038161003816119896120593(119911) (120593 (119911))10038161003816100381610038161003816 + 10038171003817100381710038171003817119862120593119879119892119896120593(119911)10038171003817100381710038171003817Z120573 lt infin

(47)

Then we have

sup119911isin119863

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1

minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le 119862 + log(43) 100381710038171003817100381710038171003817(1198921015840∘ 120593) (1205931015840)2100381710038171003817100381710038171003817V120573 lt infin

(48)

On the other hand from (15) and Lemma 4 we have

sup119899ge0

(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)120573

sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)

Hence (39) holdsThe converse implication can be shown as in the proof of

(i)

(iii) Case 1 lt 120572 lt 2 ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 hasbeen proved above We take the test function 119891120593(119911) in (23) for


every 119911 isin 119863with |120593(119911)| gt 12 by the same way as (ii) we canobtain that (40) holds


(iv) Case 120572 = 2 We have proved that (41) holds aboveTo prove (42) we consider another test function 119905119886(119911) =log(2(1 minus 119886119911)) Clearly 119905119886 isin Z2 and sup12lt|119886|lt1119905119886Z2 lt infinFor every 119911 isin 119863 with |120593(119911)| gt 12 it follows that

sup119911isin119863

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816

(50)

Applying (41) we get

sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin

(51)

Noting ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 and usingLemma 4 and (15) we conclude that (42) holds

(v) Case 120572 gt 2 We have proved that (40) holds aboveApplying test function 119891120593(119911) in (23) for every 119911 isin 119863 with|120593(119911)| gt 12 we have

1198781 = sup|120593(119911)|gt12

(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup

|120593(119911)|gt12

10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)

With the same calculation for test function 119905120593(119911)(120593(119911)) =(1 minus |120593(119911)|2)2(1 minus 120593(119911)119911)120572 with |120593(119911)| gt 12 thensup|120593(119911)|gt(12)119905120593(119911)Z120572 le 119862 and we have that

1198782 = sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2

+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1100381610038161003816100381610038161003816100381610038161003816100381610038161003816 = sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(119862120593119879119892 (119905120593(119911)))10158401015840 (120593 (119911))100381610038161003816100381610038161003816 le 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817Z120572rarrZ120573sdot sup|120593(119911)|gt12

10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin

(53)

Therefore

sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782

+ 120572 sup|120593(119911)|gt12

100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572

sdot sup|120593(119911)|gt12

100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))10038161003816100381610038161003816100381610038161003816100381610038161003816120593 (119911)10038161003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 1205721198781lt infin

(54)

Since ((1198921015840 ∘ 120593)(12059310158401015840) + (11989210158401015840 ∘ 120593)(1205931015840)2) isin 119867infinV120573 we conclude that(43) holds


operator if and only if 11989210158401205931015840 isin 119867infinV120573 and 11989210158401015840 isin 119867infinV120573 (ii) If 120572 = 1 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator

if and only if 11989210158401015840 isin 119867infinV120573 andsup119899ge0

(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin

(55)


(iii) If 1 lt 120572 lt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a boundedoperator if and only if 11989210158401015840 isin 119867infinV120573 and

sup119899ge0

(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)

(iv) If 120572 = 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if

sup119899ge0

(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0

(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin

(57)

(v) If 120572 gt 2 then 119879119892119862120593 Z120572 rarr Z120573 is a bounded operatorif and only if (56) holds and

sup119899ge0

(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)


3 Essential Norms

In this section we estimate the essential norms of theseintegral type operators on 119885119910119892119898119906119899119889 type spaces in terms of119892 120593 their derivatives and the nth power 120593119899 of 120593

Let Z120572 = 119891 isin Z120572 119891(0) = 1198911015840(0) = 0 and B120572 =119891 isin B120572 119891(0) = 0 We note that every compact operator119879 isin K(Z120572Z120573) can be extended to a compact operator119870 isinK(Z120572Z120573) In fact for every 119891 isin Z120572 119891 minus 119891(0) minus 1198911015840(0)119911 isinZ120572 and we can define119870(119891) = 119879(119891minus119891(0) minus1198911015840(0)119911) +119891(0) +1198911015840(0)119911

For 119903 isin (0 1) we consider the compact operator 119870119903 Z120572 rarr Z120573 defined by 119870119903119891(119911) = 119891(119903119911)Lemma 10 If 119883(119868119892119862120593 119862120593119868119892 119862120593119879119892 119879119892119862120593) is a bounded oper-ator fromZ120572 toZ120573 space then

119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)

Proof Clearly 119883119890Z120572rarrZ120573 ge 119883119890Z120572rarrZ120573 Then we prove119883119890Z120572rarrZ120573 le 119883119890Z120572rarrZ120573 Let 119879 isin 119870(Z120572Z120573) be given Let 119903119899 be an increasing

sequence in (0 1) converging to 1 andA = ℎ | ℎ = 119886 + 119887119911the closed subspace ofZ120572 Then

119883 minus 119879Z120572rarrZ120573 = sup119891isinZ120572119891Z120572le1

1003817100381710038171003817119883 (119891) minus 119879 (119891)1003817100381710038171003817Z120573le sup119891isinZ120572119891Z120572le1

10038171003817100381710038171003817119883 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)minus 119879 (119891 minus 119891 (0) minus 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573+ sup119891isinZ120572119891Z120572le1

10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup

119892isinZ120572119892Z120572le1

1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup

ℎisinAℎZ120572le1

1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573

(60)

Hence

inf119879isin119870(Z120572 Z120573)

119883 minus 119879Z120572rarrZ120573le inf119879isin119870(Z120572 Z120573)

1003817100381710038171003817119883 minus 119879|Z1205721003817100381710038171003817Z120572rarrZ120573+ inf119879isin119870(Z120572 Z120573)

1003817100381710038171003817119883 minus 119879|1205721003817100381710038171003817Z120572rarrZ120573le 119883119890Z120572rarrZ120573 + lim

119899rarrinfin

10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573

(61)

Since lim119899rarrinfin119883(119868minus119870119903119899)Z120572rarrZ120573 = 0 we have 119883119890Z120572rarrZ120573 le119883119890Z120572rarrZ120573 and the proof is finished

Let119863120572 Z120572 rarr B120572 and 119878120572 B120572 rarr 119867infinV120572 be the derivativeoperatorsThen clearly119863120572 and 119878120572 are linear isometries on Z120572and B120572 respectively and

119878120573119863120573119868119892119862120593119863minus1120572 119878minus1120572 = 119892 (1205931015840) 119862120593 + 1198921015840119862120593119878minus1120572 (62)

Therefore

1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890B120572rarr119867infinV120573+ 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573

(63)

Similarly

119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593

+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)


1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593 (1205931015840)2))119862120593100381710038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573

+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)

119878120573119863120573119862120593119879119892119863minus1120572 119878minus1120572= (11989210158401015840 ∘ 120593 (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593119863minus1120572 119878minus1120572

+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)

1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573

+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)

119878120573119863120573119879119892119862120593119863minus1120572 119878minus1120572 = 11989210158401015840119862120593119863minus1120572 119878minus1120572 + 1198921015840 (1205931015840) 119862120593119878minus1120572 (68)1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573le 100381710038171003817100381710038171198921015840 (1205931015840)10038171003817100381710038171003817119890B120572rarr119867infinV120573 + 100381710038171003817100381710038171198921015840101584011986212059310038171003817100381710038171003817119890Z120572rarr119867infinV120573

(69)

Theorem 11 Let 119868119892119862120593 be a bounded operator fromZ120572 toZ120573space

(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup

119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)

(ii) If 120572 = 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ maxlim sup

119899rarrinfin(119899 + 1)

sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin

(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)

(iii) If 120572 gt 1 then1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup

119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)

Proof (i) We start with the upper bound First we show that1198921015840119862120593 is a compact weighted composition operator for B120572

into 119867infinV120573 Suppose that 119891119899 is bounded sequence in B120572From Lemma 36 in [27] 119891119899 has a subsequence 119891119899119896 whichconverges uniformly on119863 to a functionwhichwe can assume

to be identically zero Then it follows from Theorem 6 andLemma 1 that

lim119896rarrinfin

sup119911isin119863

(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038161003816100381610038161003816119891119899119896 (120593 (119911))10038161003816100381610038161003816le 119862 lim119896rarrinfin

sup119911isin119863

10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)

which shows that 1198921015840119862120593 B120572 rarr 119867infinV120573 is a compact operatorand 1198921015840119862120593119890B120572rarr119867infinV120573 = 0 Applying (63) Lemmas 4 5 and 10we get that

1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 le 10038171003817100381710038171003817119892 (1205931015840) 11986212059310038171003817100381710038171003817119890119867infinV120572rarr119867infinV120573= lim sup|120593(119911)|rarr1

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816

= lim sup119899rarrinfin

10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)

120572

lim sup119899rarrinfin

(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(74)

For the lower bound let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Taking 119892119899 = 119892120593(119911119899) defined in (25)we obtain that 119892119899 is bounded sequence inZ1205720 converging to0 uniformly on compact subset of 119863 and sup119899isin119873119892119899Z120572 le 119862For every compact operator 119879 Z120572 rarr Z120573

119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge lim sup119899rarrinfin

1003817100381710038171003817100381711986811989211986212059311989211989910038171003817100381710038171003817Z120573 minus lim sup119899rarrinfin

1003817100381710038171003817119879 (119892119899)1003817100381710038171003817Z120573ge 120572 lim sup119899rarrinfin

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572

(75)

Now we use (14) and Lemma 4 to obtain that

1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817Z120572rarrZ120573ge 120572119862 lim sup|120593(119911119899)|rarr1

10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572

= 120572119862 lim sup119899rarrinfin

10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(76)

Hence (70) holds


(ii) The boundedness of 119868119892119862120593 guarantees that (1198921205931015840)119862120593 119867infinV1 rarr 119867infinV120573 and (1198921015840)119862120593 B rarr 119867infinV120573 are bounded weightedcomposition operators Theorem 34 in [28] ensures that

10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573≍ lim|120593(119911)|rarr1

(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(77)

Now we use Lemmas 4 5 and (63) to conclude that

1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 le 10038171003817100381710038171003817119892101584011986212059310038171003817100381710038171003817119890Brarr119867infinV120573 + 10038171003817100381710038171003817119892 (1205931015840)

sdot 11986212059310038171003817100381710038171003817119890119867infinV1rarr119867infinV120573 le 119862 lim sup119899rarrinfin

10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862

sdot lim sup119899rarrinfin

10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573119911119899Vlog = 119862 lim sup119899rarrinfin

(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862

sdotmaxlim sup119899rarrinfin

(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573

(78)

On the other hand let 119911119899 be a sequence in 119863 such that|120593(119911119899)| gt 12 and |120593(119911119899)| rarr 1 as 119899 rarr infin Given

ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1

minus (int1199110log3 2

1 minus 120593 (119911119899)120596119889120596)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus2

(79)

where ℎ(119911) = (119911 minus 1)((1 + log(2(1 minus 119911)))2 + 1) from [11] weknow that ℎ119899 is a bounded sequence inZ01 which convergesto zero uniformly on compact subsets of119863 and

ℎ10158401015840119899 (120593 (119911119899)) = minus120593 (119911119899)1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 ℎ1015840119899 (120593 (119911119899)) = 0sup119899

1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)

For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin By Lemmas 4 and 5 we obtainthat

119862 10038171003817100381710038171003817119868119892119862120593 minus 11987910038171003817100381710038171003817ZrarrZ120573 ge 10038171003817100381710038171003817119868119892119862120593ℎ11989910038171003817100381710038171003817ZrarrZ120573ge lim sup119899rarrinfin

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

ge lim sup119899rarrinfin

10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(81)

Now we take another function

119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)

From [11] we know that 119891119899 is a bounded sequence in Z01which converges to zero uniformly on compact subsets of119863and sup119899ge1119891119899Z lt +infin It follows from Lemmas 4 and 5 that

119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ge lim119899rarrinfin

sup 1003817100381710038171003817100381711986811989211986212059311989111989910038171003817100381710038171003817Z120573ge lim sup119899rarrinfin

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

minus lim sup119899rarrinfin

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

(83)

Noting that lim sup119899rarrinfin(119899 + 1)(1198921205931015840)120593119899V120573 le (2119862119890)119868119892119862120593119890ZrarrZ120573 we obtain(119862 + 2119862119890 ) 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890ZrarrZ120573

ge lim sup|120593(119911119899)|rarr1

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)

Hence we have 119868119892119862120593119890ZrarrZ120573 ge 119862max lim sup119899rarrinfin(119899 +1)(1198921205931015840)120593119899V120573 lim sup119899rarrinfin(log 119899)(1198921015840)120593119899V120573(iii) Let 120572 gt 1 The proof of the upper bound is similar

to that of (ii) From the proof of (i) we get that for someconstant 119862

119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup|120593(119911)|rarr1

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)

Now let 119911119899 be as before and note that the function119891119899 = 119891120593(119911119899) given in (23) Then 119891119899 is bounded sequence in


Z1205720 converging to zero uniformly on compact subsets of 119863therefore

119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ge lim119899rarrinfin

10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup

119899rarrinfin

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572minus1

100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)

By (85) we have

119862 1003817100381710038171003817100381711986811989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573ge lim sup119899rarrinfin


100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)

and the rest of the proof is similar to that of the previous andwe omit it

Theorem 12 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119862120593119879119892 Z120572 rarr Z120573 is a bounded operator(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (88)

(ii) If 120572 = 1 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573≍ lim119899rarrinfin

sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)

(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim119899rarrinfin

sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)

(iv) If 120572 = 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ≍ max lim

119899rarrinfinsup (119899 + 1)

sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim119899rarrinfin

sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(91)

(v) If 120572 gt 2 then1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z120572rarrZ120573 ≍ max lim

119899rarrinfinsup (119899 + 1)120572minus1


sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(92)

Proof (i) For the compactness of ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 theargument is similar to the proof of Theorem 11(i) then wehave ((1198921015840 ∘ 120593)(1205931015840)2)119862120593119890B120572rarr119867infinV120573 = 0 Hence by (67) andLemma 3 we get that 119862120593119879119892119890Z120572rarrZ120573 = 0

Next wewill prove (ii)The boundedness of119862120593119879119892 guaran-tees that ((1198921015840∘120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 and ((11989210158401015840∘120593)(1205931015840)2+(1198921015840∘120593)(12059310158401015840))119862120593 Z rarr 119867infinV120573 are bounded weighted compositionoperators We know that if 119906119862120593 Z rarr 119867infinV120573 is a boundedoperator then 119906119862120593 is a compact operator by Lemma 3 Hencewe consider the boundedness of 119862120593119879119892 Z rarr Z120573 and justconsider that ((1198921015840 ∘ 120593)(1205931015840)2)119862120593 B rarr 119867infinV120573 is a boundedoperator

Theorem 34 in [28] ensures that

100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2)119862120593100381710038171003817100381710038171003817119890Brarr119867infinV120573 ≍ lim|120593|rarr1

(1 minus |119911|2)120573

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(93)

From (67) and Lemmas 4 and 5 we have

1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890ZrarrZ120573 le 119862 lim sup119899rarrinfin

100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup

119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(94)

In order to prove 119862120593119879119892119890ZrarrZ120573 gelim sup119899rarrinfin(log 119899)((1198921015840 ∘ 120593)(1205931015840)2)120593119899V120573 we take the function119892119899 (119911) = 120593 (119911119899)119911 minus 1

120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)

2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 minus 119886119899(95)

where

119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (96)

and lim119899rarrinfin119886119899 = 0 From [16] we obtain that 119892119899 is abounded sequence inZ01 which converges to zero uniformlyon compact subsets of119863 By a direct calculation we have

119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)


For every compact operator 119879 Z rarr Z120573 we have119879(ℎ119899)Z120573 rarr 0 as 119899 rarr infin Let119872 = sup119899ge1119892119899Z120573 It followsfrom Lemma 5 that

11987210038171003817100381710038171003817119862120593119879119892 minus 11987910038171003817100381710038171003817119890ZrarrZ120573 ge 1003817100381710038171003817100381711986212059311987911989211989111989910038171003817100381710038171003817119890ZrarrZ120573ge lim sup119899rarrinfin

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vge lim sup119899rarrinfin

(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)

This completes the proofThe proof of (iii) is the same as that ofTheorem 11 (iii) we

do not prove it again(iv) Let 120572 = 2 Applying Lemma 3 (ii) andTheorem 32 in

[29] we get that

100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 119862120593100381710038171003817100381710038171003817119890B2rarr119867infinV120573 ≍ lim sup119899rarrinfin

(119899 + 1)sdot 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840)119862120593100381710038171003817100381710038171003817119890Z2rarr119867infinV120573≍ lim sup119899rarrinfin

(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)

which yields the upper bound by (67)With the same arguments as in the proof of Theorems 8

and 11 for some constant 119862 we have119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573


(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)

On the other hand let 119911119899 sube 119863 with |120593(119911119899)| gt 12 and|120593(119911119899)| rarr 1 as 119899 rarr infin Let the test function

119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)

From [8] we obtain that 119874119899 is a bounded sequence in Z20which converges to zero uniformly on compact subsets of119863and

lim119899rarrinfin

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816= 2 lim119899rarrinfin

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)

ApplyingTheorem 8 we get

119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 ge 10038171003817100381710038171003817119862120593119879119892 (119874119899)10038171003817100381710038171003817Z120573ge lim119899rarrinfin

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816

minus lim119899rarrinfin

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 2119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573

(103)

Hence

lim119899rarrinfin

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)

On the other hand the lower bound can be easily proved byLemmas 4 and 5

If 120572 gt 2 the proof is similar to that of (iv) except thatwe now choose the test function 119905119899(119911) = (1 minus |120593(119911119899)|2)2(1 minus120593(119911119899)119911)120572 instead of 119874119899(119911) This completes the proof ofTheorem 12

Using the same methods of Theorems 11 and 12 we canhave the following results


(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711986212059311986811989210038171003817100381710038171003817119890Z120572rarrZ120573≍ lim sup119899rarrinfin

(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)

sdot 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 lim sup119899rarrinfin

(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)

Theorem 14 Let 120593 be an analytic self-map of 119863 and 119892 isin119867(119863) and 119879119892119862120593 Z120572 rarr Z120573 is a bounded operator

(i) If 0 lt 120572 lt 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 = 0 (108)

(ii) If 120572 = 1 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890ZrarrZ120573 ≍ lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)

(iii) If 1 lt 120572 lt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ lim sup

119899rarrinfin(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (110)

(iv) If 120572 = 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z2rarrZ120573 ≍ maxlim sup

119899rarrinfin(119899 + 1)

sdot 10038171003817100381710038171003817((11989210158401205931015840)) 12059311989910038171003817100381710038171003817V120573 lim sup119899rarrinfin

(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)

(v) If 120572 gt 2 then1003817100381710038171003817100381711987911989211986212059310038171003817100381710038171003817119890Z120572rarrZ120573 ≍ maxlim sup

119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The first author was partially supported by the NationalNatural Science Foundation of China (Grant nos 1167135711571217) and the Natural Science Foundation of FujianProvince China (Grant no 2015J01005)

References

[1] R Yoneda ldquoThe composition operators on weighted Blochspacerdquo Archiv der Mathematik vol 78 no 4 pp 310ndash317 2002

[2] S Stevic ldquoOn a new operator from the logarithmic BLOch spaceto the BLOch-type space on the unit ballrdquo Applied Mathematicsand Computation vol 206 no 1 pp 313ndash320 2008

[3] S Stevic ldquoGeneralized composition operators from logarithmicBloch spaces to mixed-norm spacesrdquo Utilitas Mathematica vol77 pp 167ndash172 2008

[4] S Ye ldquoWeighted composition operators from 119865(119901 119902 119904) intologarithmic Bloch spacerdquo Journal of the Korean MathematicalSociety vol 45 no 4 pp 977ndash991 2008

[5] S Ye ldquoA weighted composition operator between differentweighted Bloch-type spacesrdquo Acta Mathematica Sinica ChineseSeries A vol 50 pp 927ndash942 2007

[6] S Ye ldquoMultipliers and cyclic vectors on the weighted BlochspacerdquoMathematical Journal of OkayamaUniversity vol 48 pp135ndash143 2006

[7] S Ye ldquoWeighted composition operator between the little 120572-Bloch spaces and the logarithmic Blochrdquo Journal of Computa-tional Analysis andApplications vol 10 no 2 pp 243ndash252 2008

[8] K Esmaeili and M Lindstrom ldquoWeighted composition opera-tors between Zygmund type spaces and their essential normsrdquoIntegral Equations and Operator Theory vol 75 no 4 pp 473ndash490 2013

[9] S Li and S Stevic ldquoWeighted composition operators fromZygmund spaces into Bloch spacesrdquo Applied Mathematics andComputation vol 206 no 2 pp 825ndash831 2008

[10] S Li and S Stevic ldquoProducts of Volterra type operator andcomposition operator from Hinfin and Bloch spaces to Zygmundspacesrdquo Journal of Mathematical Analysis and Applications vol345 no 1 pp 40ndash52 2008

[11] S Li and S Stevic ldquoGeneralized composition operators on Zyg-mund spaces and Bloch type spacesrdquo Journal of MathematicalAnalysis and Applications vol 338 no 2 pp 1282ndash1295 2008

[12] S Stevic and A K Sharma ldquoComposition operators fromweighted Bergman-Privalov spaces to Zygmund type spaces onthe unit diskrdquo Annales Polonici Mathematici vol 105 no 1 pp77ndash86 2012

[13] S Stevic ldquoOn an integral-type operator from Zygmund-typespaces to mixed-norm spaces on the unit ballrdquo Abstract andApplied Analysis vol 2010 Article ID 198608 7 pages 2010

[14] S Stevic ldquoComposition operators from the Hardy space to theZygmund-type space on the upper half-planerdquo Abstract andApplied Analysis vol 2009 Article ID 161528 8 pages 2009

[15] S Stevic ldquoOn an integral operator from the Zygmund space tothe Bloch-type space on the unit ballrdquo Glasgow MathematicalJournal vol 51 no 2 pp 275ndash287 2009

[16] S Ye and Q Hu ldquoWeighted composition operators on theZygmund spacerdquo Abstract and Applied Analysis vol 2012Article ID 462482 18 pages 2012


[17] S Ye and Z Zhuo ldquoWeighted composition operators fromHardy to Zygmund type spacesrdquo Abstract and Applied Analysisvol 2013 Article ID 365286 10 pages 2013

[18] S Ye and C Lin ldquoComposition followed by differentiation onthe Zygmund spacerdquo Acta Mathematica Sinica Chinese Seriesvol 59 no 1 pp 11ndash20 2016

[19] Z Zhuo and S Ye ldquoVolterra-type operators from analyticMorrey spaces to Bloch spacerdquo Journal of Integral Equations andApplications vol 27 no 2 pp 289ndash309 2015

[20] C Pommerenke ldquoSchlichte FUNktionen und analytischeFUNktionen von beschrankter mittlerer OszillationrdquoCommen-tarii Mathematici Helvetici vol 52 no 4 pp 591ndash602 1977

[21] E Wolf ldquoProducts of Volterra type operators and compositionoperators between weighted Bergman spaces of infinite orderand weighted BLOch type spacesrdquo Georgian MathematicalJournal vol 17 no 3 pp 621ndash627 2010

[22] S Ye ldquoProducts of volterra-type operators and compositionoperators on logarithmic Bloch spacerdquoWSEAS Transactions onMathematics vol 12 no 2 pp 180ndash188 2013

[23] S Li and S Stevic ldquoProducts of integral-type operators andcomposition operators between Bloch-type spacesrdquo Journal ofMathematical Analysis andApplications vol 349 no 2 pp 596ndash610 2009

[24] O Hyvaarinen M Kemppainen M Lindstrom A Rautio andE Saukko ldquoThe essential norm of weighted composition oper-ators on weighted Banach spaces of analytic functionsrdquo IntegralEquations and Operator Theory vol 72 no 2 pp 151ndash157 2012

[25] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society Second Series vol 61 no 3 pp872ndash884 2000

[26] OHyvaarinen andM Lindstrom ldquoEstimates of essential normsof weighted composition operators between Bloch-type spacesrdquoJournal of Mathematical Analysis and Applications vol 393 no1 pp 38ndash44 2012

[27] S Ohno K Stroethoff and R Zhao ldquoWeighted compositionoperators between Bloch-type spacesrdquo The Rocky MountainJournal of Mathematics vol 33 no 1 pp 191ndash215 2003

[28] S Stevic ldquoEssential norms of weighted composition operatorsfrom the 120572-Bloch space to a weighted-type space on the unitballrdquoAbstract andAppliedAnalysis vol 2008 Article ID 27969111 pages 2008

[29] S Stevic ldquoWeighted composition operators from Bergman-Privalov-type spaces to weighted-type spaces on the unit ballrdquoApplied Mathematics and Computation vol 217 no 5 pp 1939ndash1943 2010

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Stochastic AnalysisInternational Journal of


We notice the standard weights V120572(119911) = (1minus |119911|2)120572 where 0 lt120572 lt infin and it is well known that V120572 = V120572 We also considerthe logarithmic weight

Vlog = (log 21 minus |119911|2)minus1 119911 isin 119863 (6)

It is straightforward to show that Vlog = VlogFor an analytic self-map 120593 of119863 and a function 119906 isin 119867(119863)

we define the weighted composition operator as 119906119862120593119891 =119906 sdot (119891 ∘ 120593) for 119891 isin 119867(119863) Weighted composition operatorshave been extensively studied recently It is interesting toprovide a function theoretic characterization when 120593 and119906 induce a bounded or compact composition operator onvarious function spaces Some results on the boundednessand compactness of concrete operators between some spacesof analytic functions one of which is of Zygmund type can befound for example in [8ndash19]

Suppose that 119892 119863 rarr C is an analytic map Let 119879119892 and 119868119892denote the Volterra type operators with the analytic symbol119892 on119863 respectively

119879119892119891 (119911) = int1199110119891 (120585) 1198921015840 (120585) 119889120585 119911 isin 119863

119868119892119891 (119911) = int11991101198911015840 (120585) 119892 (120585) 119889120585 119911 isin 119863

(7)

If 119892(119911) = 119911 then 119879119892 is an integral operator While119892(119911) = ln(1(1minus119911)) then119879119892 is Cesaro operator Pommerenkeintroduced the 119881119900119897119905119890119903119903119886 type operator 119879119892 and characterizedthe boundedness of 119879119892 between 1198672 spaces in [20] Morerecently boundedness and compactness of 119881119900119897119905119890119903119903119886 typeoperators between several spaces of analytic functions havebeen studied by many authors one may see [21 22]

In this paper we consider the following integral typeoperators which were introduced by Li and Stevic (see eg[10 23]) they can be defined by

(119862120593119879119892119891) (119911) = int120593(119911)0

119891 (120585) 1198921015840 (120585) 119889120585(119862120593119868119892119891) (119911) = int120593(119911)

01198911015840 (120585) 119892 (120585) 119889120585

(119879119892119862120593119891) (119911) = int1199110119891 (120593 (120585)) 1198921015840 (120585) 119889120585

(119868119892119862120593119891) (119911) = int11991101198911015840 (120593 (120585)) 119892 (120585) 119889120585

(8)

We will characterize the boundedness of those integral typeoperators between Zygmund type spaces and also estimatetheir essential norms The boundedness and compactness ofthese operators on the logarithmic Bloch space have beencharacterized in [22]

Recall that essential norm 119879119890119883rarr119884 of a bounded linearoperator 119879 119883 rarr 119884 is defined as the distance from 119879

to K(119883 119884) the space of compact operators from 119883 to 119884namely

119879119890119883rarr119884= inf 119879 + 119870119883rarr119884 119870 119883 rarr 119884 is compact (9)

It provides a measure of noncompactness of 119879 Clearly 119879 iscompact if and only if 119879119890119883rarr119884 = 0

Throughout this paper constants are denoted by 119862 theyare positive and may differ from one occurrence to the otherThe notation 119886 ≍ 119887 means that there are positive constants1198621 1198622 such that 1198621119886 le 119887 le 11986221198862 Boundedness

In order to prove themain results of this paperWe need someauxiliary results

Lemma 1 (see [8 13]) For 0 lt 120572 lt 2 and let 119891119899 be a boundedsequence in Z120572 which converges to 0 uniformly on compactsubsets of119863 Then lim119899rarrinfinsup119911isin119863|119891119899(119911)| = 0Lemma 2 (see [8 13]) For every 119891 isin Z120572 where 120572 gt 0 onehas

(i) |1198911015840(119911)| le (2(1 minus 120572))119891Z120572 and |119891(119911)| le (2(1 minus120572))119891Z120572 for every 0 lt 120572 lt 1(ii) |1198911015840(119911)| le 2 log(2(1 minus |119911|))119891Z120572 and |119891(119911)| le 119891Z120572

for 120572 = 1(iii) |1198911015840(119911)| le (2(1 minus 120572))(119891Z120572(1 minus |119911|)120572minus1) for every120572 gt 1(iv) |119891(119911)| le (2(120572 minus 1)(2 minus 120572))119891Z120572 for every 1 lt 120572 lt 2(v) |119891(119911)| le 2 log(2(1 minus |119911|))119891Z120572 for every 120572 = 2(vi) |119891(119911)| le (2(120572 minus 1)(120572 minus 2))(119891Z120572(1 minus |119911|)120572minus2) for

every 120572 gt 2Lemma 3 (see [8]) Let 0 lt 120572 lt infin and V a radial non-increasing weight tending to 0 at boundary of 119863 and let theweighted composition operator 119906119862120593 Z120572 rarr 119867infinV be bounded

(i) If 0 lt 120572 lt 2 then 119906119862120593 is a compact operator(ii)1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890Z2rarr119867infinV ≍ lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171199061205931198991003817100381710038171003817V

≍ lim sup|120593(119911)|rarr1

V (119911) |119906 (119911)| log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(10)

(iii) If 120572 gt 2 then1003817100381710038171003817100381711990611986212059310038171003817100381710038171003817119890Z120572rarr119867infinV ≍ lim sup

119899rarrinfin(119899 + 1)120572minus2 10038171003817100381710038171199061205931198991003817100381710038171003817V

≍ lim sup|120593(119911)|rarr1

V (119911) |119906 (119911)|(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2

(11)

The following lemma is due to [24 25]




sup119899ge0

10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V ≍ sup119911isin119863

119908 (119911)V (120593 (119911)) |119906 (119911)| lt infin (12)


(ii)


10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V= lim sup|120593(119911)|rarr1

119908 (119911)V (120593 (119911)) |119906 (119911)|

(13)


lim119899rarrinfin

(119899 + 1)120572 10038171003817100381710038171199111198991003817100381710038171003817V120572 = (2120572119890 )120572 (14)

lim119899rarrinfin

(log 119899) 10038171003817100381710038171199111198991003817100381710038171003817Vlog = 1 (15)



(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (16)


sup119899ge0

(119899 + 1) 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin(17)

sup119899ge0

(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (18)


sup119899ge0

(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (19)

sup119899ge0

(119899 + 1)120572minus1 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (20)





the test functions

119891119886 (119911) = 11198862 [[


ℎ119886 (119911) = 1119886 int1199110

1 minus |119886|2(1 minus 119886119908)120572 119889119908 (24)

119892119886 (119911) = 119891119886 (119911) minus ℎ119886 (119911) (25)




100381610038161003816100381610038161003816100381610038161003816100381610038161003816120572

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(26)


Then

sup119911isin119863

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

le sup|120593(119911)|le12

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

+ sup|120593(119911)|gt12

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

le (43)120572 10038171003817100381710038171003817120593101584011989210038171003817100381710038171003817V120573 + 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 lt infin

(27)


sup119899ge0

(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (28)





100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + 119862 (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572 10038161003816100381610038161003816119868119892119862120593 (119891) (0)10038161003816100381610038161003816 = 0

100381610038161003816100381610038161003816(119868119892119862120593119891)1015840 (0)100381610038161003816100381610038161003816 = 100381610038161003816100381610038161198911015840 (120593 (0)) 119892 (0)10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572sdot 1003816100381610038161003816119892 (120593 (0))1003816100381610038161003816

(29)



119896119886 (119911) = 119901 (119886119911)119886 (log 11 minus |119886|)minus1 (30)


119901 (119911) = (119911 minus 1) ((1 + log 11 minus 119911)2 + 1) (31)



= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(32)


sup|120593(119911)|gt12

(1 minus |119911|2)120573 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ sup|120593(119911)|gt12

10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573 lt infin(33)






1003816100381610038161003816100381610038161003816100381610038161003816100381610038162120593 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(34)


sup|120593(119911)|gt12


100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le 2 sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ 2 sup|120593(119911)|gt12

10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(35)






sup119899ge0

(119899 + 1)120572 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(36)


sup119899ge0

(119899 + 1) 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162


(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin

(37)


sup119899ge0

(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(38)





sup119899ge0

(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)120573

sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (39)


sup119899ge0

(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (40)


sup119899ge0

(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(41)

sup119899ge0

(log 119899) 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log( 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162))

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(42)


sup119899ge0

(119899 + 1)120572minus2 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2

+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(43)




sup119911isin119863

(1 minus |119911|2)120573


(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)




10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)

0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816

le max|120577|le|120593(0)|

1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|


100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572

sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816

(45)





(46)


sup|120593(119911)|gt12

log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12


(47)

Then we have

sup119911isin119863

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1


(48)


sup119899ge0

(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)120573

sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)


(i)






sup119911isin119863

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816

(50)


sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin

(51)



1198781 = sup|120593(119911)|gt12

(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup

|120593(119911)|gt12

10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)


1198782 = sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2

+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))


(1 minus |119911|2)120573


10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin

(53)

Therefore

sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782

+ 120572 sup|120593(119911)|gt12

100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572

sdot sup|120593(119911)|gt12


(54)





(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin

(55)



sup119899ge0

(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)


sup119899ge0

(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0

(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin

(57)


sup119899ge0

(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)


3 Essential Norms




119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)






10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup

119892isinZ120572119892Z120572le1

1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup


1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573

(60)

Hence

inf119879isin119870(Z120572 Z120573)




119899rarrinfin

10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573

(61)




Therefore


(63)

Similarly

119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593

+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)



+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)


+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)


+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)


(69)



119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)


119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)




lim119896rarrinfin

sup119911isin119863


sup119911isin119863

10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)



(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(74)





(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572

(75)



10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(76)

Hence (70) holds




(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(77)




10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862



(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862


(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573

(78)


ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1


1 minus 120593 (119911119899)120596119889120596)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus2

(79)



1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(81)


119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

(83)


ge lim sup|120593(119911119899)|rarr1

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)




10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)





10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup

119899rarrinfin


100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)

By (85) we have



100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)




sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)


sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)


119899rarrinfinsup (119899 + 1)


sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(91)




sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(92)





(1 minus |119911|2)120573

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(93)



100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup

119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(94)


120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)

2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 minus 119886119899(95)

where

119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (96)


119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162



(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)



[29] we get that



(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)




(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)


119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)


lim119899rarrinfin


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2


(103)

Hence

lim119899rarrinfin


1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)






(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







sup119899ge0

10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V ≍ sup119911isin119863

119908 (119911)V (120593 (119911)) |119906 (119911)| lt infin (12)


(ii)


10038171003817100381710038171199061205931198991003817100381710038171003817119908119911119899V= lim sup|120593(119911)|rarr1

119908 (119911)V (120593 (119911)) |119906 (119911)|

(13)


lim119899rarrinfin

(119899 + 1)120572 10038171003817100381710038171199111198991003817100381710038171003817V120572 = (2120572119890 )120572 (14)

lim119899rarrinfin

(log 119899) 10038171003817100381710038171199111198991003817100381710038171003817Vlog = 1 (15)



(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (16)


sup119899ge0

(119899 + 1) 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin(17)

sup119899ge0

(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (18)


sup119899ge0

(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (19)

sup119899ge0

(119899 + 1)120572minus1 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 lt infin (20)





the test functions

119891119886 (119911) = 11198862 [[


ℎ119886 (119911) = 1119886 int1199110

1 minus |119886|2(1 minus 119886119908)120572 119889119908 (24)

119892119886 (119911) = 119891119886 (119911) minus ℎ119886 (119911) (25)




100381610038161003816100381610038161003816100381610038161003816100381610038161003816120572

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(26)


Then

sup119911isin119863

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

le sup|120593(119911)|le12

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

+ sup|120593(119911)|gt12

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

le (43)120572 10038171003817100381710038171003817120593101584011989210038171003817100381710038171003817V120573 + 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 lt infin

(27)


sup119899ge0

(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (28)





100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + 119862 (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572 10038161003816100381610038161003816119868119892119862120593 (119891) (0)10038161003816100381610038161003816 = 0

100381610038161003816100381610038161003816(119868119892119862120593119891)1015840 (0)100381610038161003816100381610038161003816 = 100381610038161003816100381610038161198911015840 (120593 (0)) 119892 (0)10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572sdot 1003816100381610038161003816119892 (120593 (0))1003816100381610038161003816

(29)



119896119886 (119911) = 119901 (119886119911)119886 (log 11 minus |119886|)minus1 (30)


119901 (119911) = (119911 minus 1) ((1 + log 11 minus 119911)2 + 1) (31)



= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(32)


sup|120593(119911)|gt12

(1 minus |119911|2)120573 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ sup|120593(119911)|gt12

10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573 lt infin(33)






1003816100381610038161003816100381610038161003816100381610038161003816100381610038162120593 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(34)


sup|120593(119911)|gt12


100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le 2 sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ 2 sup|120593(119911)|gt12

10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(35)






sup119899ge0

(119899 + 1)120572 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(36)


sup119899ge0

(119899 + 1) 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162


(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin

(37)


sup119899ge0

(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(38)





sup119899ge0

(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)120573

sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (39)


sup119899ge0

(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (40)


sup119899ge0

(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(41)

sup119899ge0

(log 119899) 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log( 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162))

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(42)


sup119899ge0

(119899 + 1)120572minus2 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2

+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(43)




sup119911isin119863

(1 minus |119911|2)120573


(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)




10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)

0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816

le max|120577|le|120593(0)|

1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|


100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572

sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816

(45)





(46)


sup|120593(119911)|gt12

log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12


(47)

Then we have

sup119911isin119863

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1


(48)


sup119899ge0

(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)120573

sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)


(i)






sup119911isin119863

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816

(50)


sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin

(51)



1198781 = sup|120593(119911)|gt12

(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup

|120593(119911)|gt12

10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)


1198782 = sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2

+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))


(1 minus |119911|2)120573


10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin

(53)

Therefore

sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782

+ 120572 sup|120593(119911)|gt12

100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572

sdot sup|120593(119911)|gt12


(54)





(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin

(55)



sup119899ge0

(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)


sup119899ge0

(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0

(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin

(57)


sup119899ge0

(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)


3 Essential Norms




119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)






10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup

119892isinZ120572119892Z120572le1

1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup


1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573

(60)

Hence

inf119879isin119870(Z120572 Z120573)




119899rarrinfin

10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573

(61)




Therefore


(63)

Similarly

119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593

+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)



+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)


+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)


+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)


(69)



119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)


119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)




lim119896rarrinfin

sup119911isin119863


sup119911isin119863

10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)



(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(74)





(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572

(75)



10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(76)

Hence (70) holds




(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(77)




10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862



(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862


(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573

(78)


ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1


1 minus 120593 (119911119899)120596119889120596)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus2

(79)



1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(81)


119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

(83)


ge lim sup|120593(119911119899)|rarr1

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)




10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)





10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup

119899rarrinfin


100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)

By (85) we have



100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)




sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)


sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)


119899rarrinfinsup (119899 + 1)


sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(91)




sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(92)





(1 minus |119911|2)120573

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(93)



100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup

119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(94)


120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)

2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 minus 119886119899(95)

where

119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (96)


119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162



(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)



[29] we get that



(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)




(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)


119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)


lim119899rarrinfin


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2


(103)

Hence

lim119899rarrinfin


1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)






(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





Then

sup119911isin119863

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

le sup|120593(119911)|le12

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

+ sup|120593(119911)|gt12

100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816 (1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

le (43)120572 10038171003817100381710038171003817120593101584011989210038171003817100381710038171003817V120573 + 119862 10038171003817100381710038171198921198861003817100381710038171003817Z120572 lt infin

(27)


sup119899ge0

(119899 + 1)120572 10038171003817100381710038171003817119892 (1205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (28)





100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816sdot 10038171003817100381710038171198911003817100381710038171003817Z120572 + 119862 (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 10038171003817100381710038171198911003817100381710038171003817Z120572 le 119862 10038171003817100381710038171198911003817100381710038171003817Z120572 10038161003816100381610038161003816119868119892119862120593 (119891) (0)10038161003816100381610038161003816 = 0

100381610038161003816100381610038161003816(119868119892119862120593119891)1015840 (0)100381610038161003816100381610038161003816 = 100381610038161003816100381610038161198911015840 (120593 (0)) 119892 (0)10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572sdot 1003816100381610038161003816119892 (120593 (0))1003816100381610038161003816

(29)



119896119886 (119911) = 119901 (119886119911)119886 (log 11 minus |119886|)minus1 (30)


119901 (119911) = (119911 minus 1) ((1 + log 11 minus 119911)2 + 1) (31)



= (1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

minus (1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(32)


sup|120593(119911)|gt12

(1 minus |119911|2)120573 log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

120572(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ sup|120593(119911)|gt12

10038171003817100381710038171003817119868119892119862120593 (119896120593(119911))10038171003817100381710038171003817Z120573 lt infin(33)






1003816100381610038161003816100381610038161003816100381610038161003816100381610038162120593 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(34)


sup|120593(119911)|gt12


100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816le 2 sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161205931015840 (119911) 119892 (119911)10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816100381610038161003816100381610038161003816+ 2 sup|120593(119911)|gt12

10038171003817100381710038171003817119868119892119862120593 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(35)






sup119899ge0

(119899 + 1)120572 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(36)


sup119899ge0

(119899 + 1) 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162


(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin

(37)


sup119899ge0

(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(38)





sup119899ge0

(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)120573

sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (39)


sup119899ge0

(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (40)


sup119899ge0

(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(41)

sup119899ge0

(log 119899) 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log( 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162))

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(42)


sup119899ge0

(119899 + 1)120572minus2 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2

+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(43)




sup119911isin119863

(1 minus |119911|2)120573


(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)




10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)

0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816

le max|120577|le|120593(0)|

1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|


100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572

sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816

(45)





(46)


sup|120593(119911)|gt12

log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12


(47)

Then we have

sup119911isin119863

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1


(48)


sup119899ge0

(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)120573

sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)


(i)






sup119911isin119863

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816

(50)


sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin

(51)



1198781 = sup|120593(119911)|gt12

(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup

|120593(119911)|gt12

10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)


1198782 = sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2

+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))


(1 minus |119911|2)120573


10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin

(53)

Therefore

sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782

+ 120572 sup|120593(119911)|gt12

100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572

sdot sup|120593(119911)|gt12


(54)





(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin

(55)



sup119899ge0

(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)


sup119899ge0

(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0

(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin

(57)


sup119899ge0

(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)


3 Essential Norms




119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)






10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup

119892isinZ120572119892Z120572le1

1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup


1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573

(60)

Hence

inf119879isin119870(Z120572 Z120573)




119899rarrinfin

10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573

(61)




Therefore


(63)

Similarly

119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593

+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)



+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)


+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)


+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)


(69)



119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)


119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)




lim119896rarrinfin

sup119911isin119863


sup119911isin119863

10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)



(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(74)





(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572

(75)



10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(76)

Hence (70) holds




(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(77)




10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862



(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862


(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573

(78)


ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1


1 minus 120593 (119911119899)120596119889120596)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus2

(79)



1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(81)


119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

(83)


ge lim sup|120593(119911119899)|rarr1

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)




10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)





10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup

119899rarrinfin


100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)

By (85) we have



100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)




sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)


sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)


119899rarrinfinsup (119899 + 1)


sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(91)




sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(92)





(1 minus |119911|2)120573

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(93)



100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup

119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(94)


120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)

2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 minus 119886119899(95)

where

119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (96)


119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162



(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)



[29] we get that



(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)




(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)


119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)


lim119899rarrinfin


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2


(103)

Hence

lim119899rarrinfin


1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)






(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








sup119899ge0

(119899 + 1)120572 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

100381610038161003816100381610038161003816119892 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(36)


sup119899ge0

(119899 + 1) 100381710038171003817100381710038171003817(119892 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162


(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin

(37)


sup119899ge0

(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2 + 119892 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(38)





sup119899ge0

(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)120573

sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (39)


sup119899ge0

(119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin (40)


sup119899ge0

(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(41)

sup119899ge0

(log 119899) 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log( 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162))

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2 + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(42)


sup119899ge0

(119899 + 1)120572minus2 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573≍ sup119911isin119863


10038161003816100381610038161003816100381611989210158401015840 (120593 (119911)) (1205931015840 (119911))2

+ 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816 lt infin(43)




sup119911isin119863

(1 minus |119911|2)120573


(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)




10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)

0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816

le max|120577|le|120593(0)|

1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|


100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572

sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816

(45)





(46)


sup|120593(119911)|gt12

log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12


(47)

Then we have

sup119911isin119863

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1


(48)


sup119899ge0

(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)120573

sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)


(i)






sup119911isin119863

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816

(50)


sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin

(51)



1198781 = sup|120593(119911)|gt12

(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup

|120593(119911)|gt12

10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)


1198782 = sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2

+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))


(1 minus |119911|2)120573


10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin

(53)

Therefore

sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782

+ 120572 sup|120593(119911)|gt12

100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572

sdot sup|120593(119911)|gt12


(54)





(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin

(55)



sup119899ge0

(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)


sup119899ge0

(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0

(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin

(57)


sup119899ge0

(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)


3 Essential Norms




119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)






10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup

119892isinZ120572119892Z120572le1

1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup


1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573

(60)

Hence

inf119879isin119870(Z120572 Z120573)




119899rarrinfin

10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573

(61)




Therefore


(63)

Similarly

119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593

+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)



+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)


+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)


+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)


(69)



119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)


119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)




lim119896rarrinfin

sup119911isin119863


sup119911isin119863

10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)



(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(74)





(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572

(75)



10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(76)

Hence (70) holds




(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(77)




10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862



(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862


(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573

(78)


ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1


1 minus 120593 (119911119899)120596119889120596)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus2

(79)



1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(81)


119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

(83)


ge lim sup|120593(119911119899)|rarr1

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)




10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)





10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup

119899rarrinfin


100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)

By (85) we have



100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)




sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)


sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)


119899rarrinfinsup (119899 + 1)


sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(91)




sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(92)





(1 minus |119911|2)120573

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(93)



100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup

119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(94)


120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)

2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 minus 119886119899(95)

where

119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (96)


119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162



(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)



[29] we get that



(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)




(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)


119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)


lim119899rarrinfin


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2


(103)

Hence

lim119899rarrinfin


1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)






(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







sup119911isin119863

(1 minus |119911|2)120573


(1 minus |119911|2)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 lt infin(44)




10038161003816100381610038161003816119862120593119879119892 (119891) (0)10038161003816100381610038161003816 =100381610038161003816100381610038161003816100381610038161003816int120593(0)

0119891 (120577) 1198921015840 (120577) 119889120577100381610038161003816100381610038161003816100381610038161003816

le max|120577|le|120593(0)|

1003816100381610038161003816119891 (120577)1003816100381610038161003816 max|120577|le|120593(0)|


100381610038161003816100381610038161198921015840 (120577)10038161003816100381610038161003816 100381610038161003816100381610038161003816(119862120593119879119892119891)1015840 (0)100381610038161003816100381610038161003816 = 10038161003816100381610038161003816119891 (120593 (0)) 1205931015840 (0) 1198921015840 (120593 (0))10038161003816100381610038161003816 le 10038171003817100381710038171198911003817100381710038171003817Z120572

sdot 100381610038161003816100381610038161205931015840 (0)10038161003816100381610038161003816 100381610038161003816100381610038161198921015840 (120593 (0))10038161003816100381610038161003816

(45)





(46)


sup|120593(119911)|gt12

log 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162 (1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 le sup|120593(119911)|gt12


(47)

Then we have

sup119911isin119863

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816le sup|120593(119911)|le12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2

sdot 1198921015840 (120593 (119911))100381610038161003816100381610038161003816 + sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1


(48)


sup119899ge0

(log 119899) 100381710038171003817100381710038171003817(1198921015840 ∘ 120593) (1205931015840)2 120593119899100381710038171003817100381710038171003817V120573 ≍ sup119911isin119863

(1 minus |119911|2)120573

sdot log( 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 (49)


(i)






sup119911isin119863

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816

(50)


sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin

(51)



1198781 = sup|120593(119911)|gt12

(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup

|120593(119911)|gt12

10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)


1198782 = sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2

+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))


(1 minus |119911|2)120573


10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin

(53)

Therefore

sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782

+ 120572 sup|120593(119911)|gt12

100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572

sdot sup|120593(119911)|gt12


(54)





(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin

(55)



sup119899ge0

(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)


sup119899ge0

(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0

(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin

(57)


sup119899ge0

(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)


3 Essential Norms




119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)






10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup

119892isinZ120572119892Z120572le1

1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup


1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573

(60)

Hence

inf119879isin119870(Z120572 Z120573)




119899rarrinfin

10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573

(61)




Therefore


(63)

Similarly

119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593

+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)



+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)


+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)


+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)


(69)



119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)


119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)




lim119896rarrinfin

sup119911isin119863


sup119911isin119863

10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)



(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(74)





(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572

(75)



10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(76)

Hence (70) holds




(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(77)




10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862



(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862


(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573

(78)


ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1


1 minus 120593 (119911119899)120596119889120596)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus2

(79)



1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(81)


119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

(83)


ge lim sup|120593(119911119899)|rarr1

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)




10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)





10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup

119899rarrinfin


100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)

By (85) we have



100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)




sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)


sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)


119899rarrinfinsup (119899 + 1)


sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(91)




sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(92)





(1 minus |119911|2)120573

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(93)



100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup

119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(94)


120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)

2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 minus 119886119899(95)

where

119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (96)


119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162



(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)



[29] we get that



(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)




(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)


119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)


lim119899rarrinfin


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2


(103)

Hence

lim119899rarrinfin


1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)






(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of








sup119911isin119863

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(119862120593119879119892119905119886)10158401015840 (119911)100381610038161003816100381610038161003816ge sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816minus sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816

(50)


sup|120593(119911)|gt12

log( 11 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)(1 minus |119911|2)120573

sdot 100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)100381610038161003816100381610038161003816le sup|120593(119911)|gt12

(1 minus |119911|2)1205731 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (120593 (119911)) (120593 (119911))210038161003816100381610038161003816+ 1003817100381710038171003817100381711986212059311987911989211990511988610038171003817100381710038171003817Z120573 lt infin

(51)



1198781 = sup|120593(119911)|gt12

(1 minus |119911|2)120573100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(1205931015840 (119911))2 1198921015840 (120593 (119911))120593 (119911) (1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1

100381610038161003816100381610038161003816100381610038161003816100381610038161003816= sup|120593(119911)|gt12

(1 minus |119911|2)120573 100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198911015840120593(119911) (120593 (119911))10038161003816100381610038161003816 le sup

|120593(119911)|gt12

10038171003817100381710038171003817119862120593119879119892 (119891120593(119911))10038171003817100381710038171003817Z120573 lt infin(52)


1198782 = sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2

+ 120572120593 (119911)(1205931015840 (119911))2 1198921015840 (120593 (119911))


(1 minus |119911|2)120573


10038171003817100381710038171003817119905120593(119911)10038171003817100381710038171003817Z120572 lt infin

(53)

Therefore

sup|120593(119911)|gt12

(1 minus |119911|2)120573

sdot100381610038161003816100381610038161003816100381610038161003816100381610038161003816(1205931015840 (119911))2 11989210158401015840 (120593 (119911)) + 1198921015840 (120593 (119911)) 12059310158401015840 (119911)

(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus2100381610038161003816100381610038161003816100381610038161003816100381610038161003816 le 1198782

+ 120572 sup|120593(119911)|gt12

100381610038161003816100381610038161003816(1205931015840 (119911))2 1198921015840 (120593 (119911))100381610038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572minus1 le 1198782 + 120572

sdot sup|120593(119911)|gt12


(54)





(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816lt infin

(55)



sup119899ge0

(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)


sup119899ge0

(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0

(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin

(57)


sup119899ge0

(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)


3 Essential Norms




119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)






10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup

119892isinZ120572119892Z120572le1

1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup


1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573

(60)

Hence

inf119879isin119870(Z120572 Z120573)




119899rarrinfin

10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573

(61)




Therefore


(63)

Similarly

119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593

+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)



+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)


+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)


+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)


(69)



119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)


119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)




lim119896rarrinfin

sup119911isin119863


sup119911isin119863

10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)



(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(74)





(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572

(75)



10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(76)

Hence (70) holds




(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(77)




10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862



(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862


(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573

(78)


ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1


1 minus 120593 (119911119899)120596119889120596)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus2

(79)



1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(81)


119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

(83)


ge lim sup|120593(119911119899)|rarr1

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)




10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)





10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup

119899rarrinfin


100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)

By (85) we have



100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)




sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)


sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)


119899rarrinfinsup (119899 + 1)


sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(91)




sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(92)





(1 minus |119911|2)120573

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(93)



100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup

119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(94)


120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)

2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 minus 119886119899(95)

where

119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (96)


119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162



(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)



[29] we get that



(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)




(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)


119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)


lim119899rarrinfin


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2


(103)

Hence

lim119899rarrinfin


1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)






(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






sup119899ge0

(119899 + 1)120572minus1 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infin (56)


sup119899ge0

(119899 + 1) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

100381610038161003816100381610038161198921015840 (119911) 1205931015840 (119911)10038161003816100381610038161003816 lt infinsup119899ge0

(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863

(1 minus |119911|2)120573 log 2(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)

1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin

(57)


sup119899ge0

(119899 + 1)120572minus2 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573≍ sup119911isin119863


1003816100381610038161003816100381611989210158401015840 (119911)10038161003816100381610038161003816 lt infin (58)


3 Essential Norms




119883119890Z120572rarrZ120573 = 119883119890Z120572rarrZ120573 (59)






10038171003817100381710038171003817119883 (119891 (0) + 1198911015840 (0) 119911)minus 119879 (119891 (0) + 1198911015840 (0) 119911)10038171003817100381710038171003817Z120573 le sup

119892isinZ120572119892Z120572le1

1003817100381710038171003817119883 (119892)minus 119879|Z120572 (119892)1003817100381710038171003817Z120573 + sup


1003817100381710038171003817119883 (ℎ) minus 119879|120572 (ℎ)1003817100381710038171003817Z120573

(60)

Hence

inf119879isin119870(Z120572 Z120573)




119899rarrinfin

10038171003817100381710038171003817119883 (119868 minus 119870119903119899)10038171003817100381710038171003817Z120572rarrZ120573

(61)




Therefore


(63)

Similarly

119878120573119863120573119862120593119868119892119863minus1120572 119878minus1120572= (119892 ∘ 120593 (1205931015840)2)119862120593

+ (1198921015840 ∘ 120593 (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593119878minus1120572 (64)



+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)


+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)


+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)


(69)



119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)


119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)




lim119896rarrinfin

sup119911isin119863


sup119911isin119863

10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)



(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(74)





(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572

(75)



10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(76)

Hence (70) holds




(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(77)




10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862



(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862


(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573

(78)


ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1


1 minus 120593 (119911119899)120596119889120596)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus2

(79)



1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(81)


119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

(83)


ge lim sup|120593(119911119899)|rarr1

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)




10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)





10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup

119899rarrinfin


100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)

By (85) we have



100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)




sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)


sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)


119899rarrinfinsup (119899 + 1)


sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(91)




sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(92)





(1 minus |119911|2)120573

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(93)



100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup

119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(94)


120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)

2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 minus 119886119899(95)

where

119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (96)


119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162



(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)



[29] we get that



(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)




(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)


119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)


lim119899rarrinfin


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2


(103)

Hence

lim119899rarrinfin


1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)






(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






+ 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + 119892 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890B120572rarr119867infinV120573 (65)


+ (1198921015840 ∘ 120593 (1205931015840)2)119862120593119878minus1120572 (66)


+ 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + 1198921015840 ∘ 120593 (12059310158401015840))119862120593100381710038171003817100381710038171003817119890Z120572rarr119867infinV120573 (67)


(69)



119899rarrinfin(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 (70)


119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (71)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (72)




lim119896rarrinfin

sup119911isin119863


sup119911isin119863

10038161003816100381610038161003816119891119899119896 (119911)10038161003816100381610038161003816 = 0 (73)



(1 minus |119911|2)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572

10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= ( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(74)





(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)12057310038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572

(75)



10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V120572= 119862( 1198902120572)

120572


(119899 + 1)120572 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(76)

Hence (70) holds




(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(77)




10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862



(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862


(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573

(78)


ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1


1 minus 120593 (119911119899)120596119889120596)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus2

(79)



1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(81)


119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

(83)


ge lim sup|120593(119911119899)|rarr1

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)




10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)





10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup

119899rarrinfin


100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)

By (85) we have



100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)




sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)


sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)


119899rarrinfinsup (119899 + 1)


sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(91)




sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(92)





(1 minus |119911|2)120573

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(93)



100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup

119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(94)


120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)

2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 minus 119886119899(95)

where

119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (96)


119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162



(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)



[29] we get that



(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)




(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)


119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)


lim119899rarrinfin


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2


(103)

Hence

lim119899rarrinfin


1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)






(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







(1 minus |119911|2)120573 100381610038161003816100381610038161198921015840 (119911)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(77)




10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1 + 119862



(log 119899) 10038171003817100381710038171003817119892101584012059311989910038171003817100381710038171003817V120573+ 1198621198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 le 119862


(119899 + 1)sdot 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573 lim sup

119899rarrinfin(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573

(78)


ℎ119899 (119911) = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1


1 minus 120593 (119911119899)120596119889120596)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus2

(79)



1003817100381710038171003817ℎ1198991003817100381710038171003817Z lt +infin(80)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573119911119899V1= 1198902 lim sup

119899rarrinfin(119899 + 1) 10038171003817100381710038171003817(1198921205931015840) 12059311989910038171003817100381710038171003817V120573

(81)


119891119899 = ℎ (120593 (119911119899)119911)120593 (119911119899) (log 2

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)minus1 (82)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)10038161003816100381610038161 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162

(83)


ge lim sup|120593(119911119899)|rarr1

(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162


(log 119899) 10038171003817100381710038171003817(1198921015840) 12059311989910038171003817100381710038171003817V120573 (84)




10038161003816100381610038161003816119892 (119911) 1205931015840 (119911)10038161003816100381610038161003816 (1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573(1 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162)120572 (85)





10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup

119899rarrinfin


100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)

By (85) we have



100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)




sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)


sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)


119899rarrinfinsup (119899 + 1)


sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(91)




sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(92)





(1 minus |119911|2)120573

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(93)



100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup

119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(94)


120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)

2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 minus 119886119899(95)

where

119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (96)


119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162



(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)



[29] we get that



(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)




(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)


119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)


lim119899rarrinfin


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2


(103)

Hence

lim119899rarrinfin


1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)






(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







10038171003817100381710038171003817119868119892119862120593 (119891119899)10038171003817100381710038171003817Z120573ge 2120572 lim sup

119899rarrinfin


100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 10038161003816100381610038161003816119892 (119911119899) 1205931015840 (119911119899)100381610038161003816100381610038161003816100381610038161003816120593 (119911119899)1003816100381610038161003816 (1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)120572 (86)

By (85) we have



100381610038161003816100381610038161198921015840 (119911119899)10038161003816100381610038161003816 (87)




sup (log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (89)


sup (119899 + 1)120572minus1 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (90)


119899rarrinfinsup (119899 + 1)


sup (log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(91)




sup (119899 + 1)120572minus2sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(92)





(1 minus |119911|2)120573

sdot 1003816100381610038161003816100381610038161198921015840 (120593 (119911)) (1205931015840 (119911))2100381610038161003816100381610038161003816 log 21 minus 1003816100381610038161003816120593 (119911)10038161003816100381610038162

(93)



100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573119911119899Vlog= 119862 lim sup

119899rarrinfin(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(94)


120593 (119911119899) ((1 + log 11 minus 120593 (119911119899)119911)

2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 minus 119886119899(95)

where

119886119899 = 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 minus 1120593 (119911119899) ((1 + log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)2 + 1)

sdot (log 11 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (96)


119892119899 (120593 (119911119899)) = 01198921015840119899 (120593 (119911119899)) = log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 (97)




(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162



(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)



[29] we get that



(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)




(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)


119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)


lim119899rarrinfin


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2


(103)

Hence

lim119899rarrinfin


1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)






(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot log 1

1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162



(log 119899) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573

(98)



[29] we get that



(log 119899)sdot 100381710038171003817100381710038171003817((11989210158401015840 ∘ 120593) (1205931015840)2 + (1198921015840 ∘ 120593) 12059310158401015840) 120593119899100381710038171003817100381710038171003817V120573

(99)




(119899 + 1) 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (100)


119874119899 (119911) = (1 + (log 21 minus 120593 (119911119899)119911)

2)

sdot (log 21 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162)

minus1 (101)


lim119899rarrinfin


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)1205731 minus 1003816100381610038161003816120593 (119911119899)100381610038161003816100381621003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816

sdot 1003816100381610038161003816120593 (119911119899)1003816100381610038161003816 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (102)



(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot 1003816100381610038161003816119874119899 (120593 (119911119899))1003816100381610038161003816


(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573 1003816100381610038161003816100381610038161198921015840 (120593 (119911119899)) (1205931015840 (119911119899))2100381610038161003816100381610038161003816sdot 100381610038161003816100381610038161198741015840119899 (120593 (119911119899))10038161003816100381610038161003816 ge lim

119899rarrinfin(1 minus 100381610038161003816100381611991111989910038161003816100381610038162)120573

sdot 10038161003816100381610038161003816100381611989210158401015840 (120593 (119911119899)) (1205931015840 (119911119899))2 + 1198921015840 (120593 (119911119899)) 12059310158401015840 (119911119899)100381610038161003816100381610038161003816sdot log 2


(103)

Hence

lim119899rarrinfin


1 minus 1003816100381610038161003816120593 (119911119899)10038161003816100381610038162 le 119862 1003817100381710038171003817100381711986212059311987911989210038171003817100381710038171003817119890Z2rarrZ120573 (104)






(119899 + 1)120572 100381710038171003817100381710038171003817((119892 ∘ 120593) (1205931015840)2) 120593119899100381710038171003817100381710038171003817V120573 (105)



119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119899rarrinfin(119899 + 1)


(log 119899)sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(106)


119899rarrinfin(119899 + 1)120572


(119899 + 1)120572minus1sdot 100381710038171003817100381710038171003817((1198921015840 ∘ 120593) (1205931015840)2 + (119892 ∘ 120593) (12059310158401015840)) 120593119899100381710038171003817100381710038171003817V120573

(107)




119899rarrinfin(log 119899) 10038171003817100381710038171003817(11989210158401205931015840) 12059311989910038171003817100381710038171003817V120573 (109)




119899rarrinfin(119899 + 1)


(log 119899) 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573 (111)


119899rarrinfin(119899 + 1)120572minus1


(119899 + 1)120572minus2sdot 10038171003817100381710038171003817(11989210158401015840) 12059311989910038171003817100381710038171003817V120573

(112)



Acknowledgments


References






































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




essential norms of volterra type operators between zygmund

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