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SHARP WEIGHTED ESTIMATES FOR MULTILINEAR

CALDERON-ZYGMUND OPERATORS ON NON-HOMOGENEOUS

SPACES

ABHISHEK GHOSH, ANKIT BHOJAK, PARASAR MOHANTY AND SAURABHSHRIVASTAVA

ABSTRACT. In this article, we address pointwise sparse domination formultilinear Calderon-Zygmund operators on upper doubling, geometri-cally doubling metric measure spaces. As a consequence, we have ob-tained sharp quantitative weighted estimates for multilinear Calderon-Zygmund operators.

1. INTRODUCTION

Pointwise domination of Calderon-Zygmund operators by some specialdiscrete operators (called as “sparse operators”) was initiated by the worksof Conde-Alonso and Rey in [2], Lerner and Nazarov in [13]. In [10], Laceydeveloped an innovative approach to dominate Calderon-Zygmund op-erators by sparse operators and relaxed the log-Dini condition to standardDini condition. Subsequently, Lerner obtained an elegant proof of the point-wise domination of Calderon-Zygmund operators by sparse operators in[12]. In [5], Grafakos and Torres initiated the study of multilinear Calderon-Zygmund operators on Rn. Later in [6], they have shown that the mul-tilinear Calderon-Zygmund operator can be controlled by the product ofHardy-Littlewood maximal functions in terms of norms and the authorshave concluded some weighted boundedness of the multilinear Calderon-Zygmund operators in terms of linear Muckenhoupt classes. The ques-tion of controlling a multilinear Calderon-Zygmund operator by a smallermaximal operator and developing the multilinear weighted theory is ad-dressed by Lerner et al in [11]. Later its sharp quantitative dependence onA~P

characteristic was addressed in [14] by adopting recently developedsparse domination techniques in the multilinear settings. In [3], point-wise sparse domination for multilinear Calderon-Zygmund operators wasachieved in Euclidean setting. Our goal in this article is to address the issueof dominating multilinear Calderon-Zygmund operators by sparse opera-tors in the non-homogeneous setting and to achieve quantitative weightedestimates. Our domination of the multilinear Calderon-Zygmund opera-tor will be pointwise. To state our results in details we recall the followingpreliminaries.

In [19, 20], X. Tolsa established the existence of the principal value ofthe Cauchy integral operator on C with respect to arbitrary positive Radon

1991 Mathematics Subject Classification. Primary 42B25; Secondary 42B20.Key words and phrases. Multilinear Calderon-Zygmund operator, Weights, Upper dou-

bling measure, Geometrically doubling condition.

1

http://arxiv.org/abs/1609.08995v3

2 A. GHOSH, A. BHOJAK, P. MOHANTY AND S. SHRIVASTAVA

measures µ satisfying µ(Dr) ≤ Cr where Dr represents a disc with ra-dius r and C is a constant independent of r. These works of Tolsa playeda deep role in proving the celebrated Painleve’s problem ([20, 22, 23]) aswell as a prelude to the study of Calderon-Zygmund operators on non-homogeneous spaces. Nazarov, Treil, and Volberg ([16, 17, 18]) have de-veloped an innovative approach to treat Calderon-Zygmund operators onnon-homogeneous spaces. They obtained the weak end-point result fromthe estimates on point masses and Whitney decomposition. Tolsa in [22, 23]developed a suitable Calderon-Zygmund decomposition on Rn with non-doubling measures of polynomial growth, as a consequence, he obtainedthe weak type end-point boundedness of Calderon-Zygmund operators onnon-homogeneous spaces (see [21]). T. Hytonen, in [8], provided a uni-fied framework to study Calderon-Zygmund operators in upper doubling,geometric doubling metric measure spaces. Throughout this article we re-fer an upper doubling, geometric doubling metric measure space as “non-homogeneous” space. We start by defining upper doubling measures.

Definition 1.1 ([8, 24]). A metric measure space (X, d, µ) is upper doubling ifthere exists a dominating function λ : X × (0,∞) → (0,∞) and a constantCλ > 0 such that

i) µ(B(x, r)) ≤ λ(x, r) for all x ∈ X, r > 0, where B(x, r) denotes the ball ofradius r with center at x ∈ X.

ii) For every x ∈ X the function r → λ(x, r) is non-decreasing.iii) λ satisfies the doubling property: λ(x, r) ≤ Cλλ(x,

r2 ) for all x ∈ X, r > 0.

In particular, if λ(x, r) = Crd for some d > 0 and constant C > 0 then we say µhas polynomial growth.

Definition 1.2 ([24]). A metric space (X, d) is called geometric doubling(withdoubling dimension N ) if any ball B(x, r) ⊂ X can contain at most N centres{yi} of disjoint balls {B(yi, r/4)}

Ni=1.

Recently in [24], A. Volberg and P. Zorin-Kranich used the sparse dom-ination to prove the appropriate weighted boundedness of the Calderon-Zygmund operators on non-homogeneous spaces. In this paper, we willaddress the pointwise domination of multilinear Calderon-Zygmund op-erators by appropriate sparse operators on non-homogeneous spaces. Theweak type weighted boundedness of multilinear Calderon-Zygmund oper-ators on non-homogeneous spaces has been considered in [7]. In this work,they have assumed the unweighted weak type end-point boundedness formultilinear Calderon-Zygmund operators.

The goal of this article is twofold, first, we prove the natural end-pointboundedness of multilinear Calderon-Zygmund operators on non-homogeneousspaces. Secondly, we provide appropriate sparse domination results andquantitative weighted estimates. Using sparse domination techniques wehave obtained strong (in contrast to weak type bound in [7]) weightedboundedness for multilinear Calderon-Zygmund operators. Besides, as wedominate multilinear Calderon-Zygmund operators pointwise by sparseoperators, our result addresses more exponents than given in [14]. Ourbound is also sharp (when restricted to Euclidean setting) and holds witha much weaker assumption than [7]. Also, we have proved an analogue of

MULTILINEAR CALDERON-ZYGMUND OPERATORS ON NON-HOMOGENEOUS SPACES 3

Cotlar’s type inequality for multilinear maximal Calderon-Zygmund oper-ators in this setting. The end-point boundedness of multilinear Calderon-Zygmund operators on Rn for polynomial growth measures was addressed

in [15]. Throughout this article ~P will denote the tuple ~P = (p1, p2) with1 ≤ p1, p2 < ∞ and 1

p= 1

p1+ 1

p2. Given a tuple of weights ~w = (w1, w2) and

a tuple ~P = (p1, p2) we set v~w =∏2

j=1w

ppj

j . A . B is the abbreviation forA ≤ CB, where C is independent of A and B.

2. DEFINITIONS AND RESULTS

We start by defining multilinear Calderon-Zygmund operators on non-homogeneous spaces.

Definition 2.1. Let (X, d, µ) be a non-homogeneous space. A Calderon-Zygmundkernel K is µ-locally integrable function defined away from the diagonal x = y1 =y2 in X3 and satisfying the following:Size condition: There exist a constant CK > 0 such that whenever x 6= yj forsome j = 1, 2,

|K(x, y1, y2)| ≤ CK1

(λ(x, d(x, y1)) + λ(x, d(x, y2)))2.(1)

Regularity conditions: There exist a constant CK > 0 such that the followingregularity conditions hold.

|K(x, y1, y2)−K(x′, y1, y2)| ≤ CK1

(λ(x, d(x, y1)) + λ(x, d(x, y2)))2ω

d(x,x′)2∑

i=1d(x,yi)

,

(2)

whenever d(x, x′) ≤ 12 maxi=1,2

d(x, yi). Also similar regularity conditions hold for

other components. ω is a modulus of continuity satisfying the Dini(1) conditioni.e., ω is a monotonically increasing subadditive function with w(0) = 0 and1∫

0

ω(t)tdt = ||ω||Dini(1) < ∞.

Definition 2.2. We say that an operator T is a bilinear Calderon-Zygmund op-erator if it is bounded from Lp1(X, d, µ) × Lp2(X, d, µ) into Lp(X, d, µ) where

1p=

2∑

i=1

1pi, 1 < p, p1, p2 < ∞ and there exists a Calderon-Zygmund kernel K

such that T is represented for compactly supported bounded functions as,

(3) T (f1, f2)(x) =

∫

X2

K(x, y1, y2)f1(y1)f2(y2)dµ(y1)dµ(y2)

for a.e x /∈2⋂

j=1supp(fj). Corresponding to a bilinear Calderon-Zygmund operator

T , we define the bilinear maximal truncated operator by

(4) T ∗(f1, f2)(x) = supǫ>0

|Tǫ(f1, f2)(x)|,


where Tǫ is defined as following,

Tǫ(f1, f2)(x) =

∫

2∑

i=1d(x,yi)2>ǫ2

K(x, y1, y2)f1(y1)f2(y2)dµ(y1)dµ(y2).

We would like to mention that our definition does not demand the weak-type end point estimate like [7]. We require another form of the truncatedmaximal function for geometric compatibility and it is easier to work withthe following truncated maximal operator (see [6] for more details).

T ∗(f1, f2)(x) = supǫ>0

|Tǫ(f1, f2)(x)|,

where,

Tǫ(f1, f2)(x) :=

∫

maxi=1,2

d(x,yi)>ǫ

K(x, y1, y2)f1(y1)f2(y2)dµ(~y).

The following remark says the truncated maximal operators are equivalentupto a maximal function.

Remark 2.3. Denote Sǫ(x) = {~y ∈ X2 : maxi=1,2

d(x, yi) ≤ ǫ} and Uǫ(x) := {~y ∈

Sǫ(x) : d(x, y1)2 + d(x, y2)

2 ≥ ǫ2}. It is very easy to observe

|Tǫ(f1, f2)(x)− Tǫ(f1, f2)(x)| ≤

∫

Uǫ(x)|K(x, y1, y2)||f1(y1)||f2(y2)|dµ(y1)dµ(y2)

≤ CK,λMλ(f1, f2)(x),

where,

Mλ(f1, f2)(x) := supr>0

1

λ(x, r)2

∫

B(x,r)|f1(y1)|dµ(y1)

∫

B(x,r)|f2(y2)|dµ(y2).

(5)

Therefore, T ∗(f1, f2) and T ∗(f1, f2) are pointwise a.e. equivalent upto the maxi-mal function Mλ. The maximal operator Mλ plays a crucial role similar to thatof Hardy-Littlewood maximal function in [12]. It is easy to verify that Mλ is

bounded from L1(X, d, µ) × L1(X, d, µ) to L12,∞(X, d, µ) as Mλ is bounded

by M cµ(f1)(x)M

cµ(f2)(x) pointwise and M c

µ is weak-type (1,1) with respect toµ, where M c

µ is the centred Hardy-Littlewood maximal function with respect to µ.

Now we state the main sparse domination principle of this article. Ourstrategy of sparse domination for multilinear Calderon-Zygmund operatoris motivated by the work of Volberg and Kranich in [24]. In the absenceof a compatible dyadic structure the authors in [24] have used the David-Mattila cells. We also need them for our purpose and a detailed descriptionof David-Mattila cells is given in Lemma 3.1 (see also [4, 24]). D will denotethe collection of David-Mattila cells and corresponding to each cell Q ∈ D ,B(Q) will denote the associated ball with respect to Q satisfying certainproperties. We recall the definition of a sparse family in this setting.

Definition 2.4. A family of David-Mattila cells S ⊂ D is said to be η-sparse,0 < η < 1, if for every Q ∈ S there exists a measurable set EQ ⊂ Q such that


i) µ(EQ) ≥ ηµ(Q) andii) the sets {EQ}Q∈S are pairwise disjoint.

Given a sparse family S in X and a large number α ≥ 200, the bilinear sparseoperator is defined by

AS(~f)(x) =∑

Q∈S

(

2∏

i=1

1

µ(αB(Q))

∫

30B(Q)|fi|dµ

)

χQ.

Now we are in a position to state the main sparse domination principle.

Theorem 2.5. Let T be a bilinear Calderon-Zygmund operator defined on a non-homogeneous space (X, d, µ). Let α ≥ 200 . Then for every bounded subset X ′ ofX and for integrable functions fj , with supp(fj) ⊆ X ′ for j = 1, 2, there existsparse families Gn, n ≥ 0, of David-Mattila cells such that the sparse domination

T ∗(~f)(x) .T,α

∞∑

n=0

100−nAGn(~f)(x)(6)

holds pointwise a.e. on X ′.

In Proposition 3.8, we will provide quantitative weighted boundednessof bilinear sparse operators. As a consequence of Theorem 2.5 and Proposi-tion 3.8, we obtain the following corollary concluding the weighted bound-edness of bilinear Calderon-Zygmund operators on non-homogeneous spaces.

Corollary 2.6. Let T be a bilinear Calderon-Zygmund operator defined on a non-homogeneous space (X, d, µ). Then for all exponents 1 < p1, p2 < ∞ satisfying

1p=

2∑

i=1

1pi

and any multiple weight ~w, we have

‖T ∗(~f)‖Lp(v~w) . Cw

2∏

i=1

‖fi‖Lpi (wi),

where the implicit constant is independent of ~w, and

Cw =

supQ

v~w(Q)p′0p

2∏

i=1σi(αB(Q))

p′0p′i

µ(αB(Q))2µ(Q)2(p′0−1)

if 12 < p ≤ 1 and p0 = min

ipi

supQ

v~w(Q)2∏

i=1σi(αB(Q))

p

p′i

µ(αB(Q))2µ(Q)2(p−1) if p ≥ maxi

p′i

supQ

v~w(Q)

p′jp σj(αB(Q))

2∏

i=1,i6=j

σi(αB(Q))

p′jp′i

µ(αB(Q))2µ(Q)2(p′

j−1)

if p′j = max{p, p′1, p′2}.

Remark 2.7. Let the underlying measure be a doubling measure and ~w ∈ A~P(µ)

i.e.,

[~w]A~P:= sup

B

(

1

µ(B)

∫

B

v~wdµ

) 2∏

j=1

(

1

µ(B)

∫

B

w1−p′jj dµ

)p

p′j≤ K < ∞.


As in this case µ(αB(Q)) and µ(B(Q)) are comparable thus in each of the cases

mentioned above, we get Cw ≃ [~w]max(1,

p′1p,p′2p)

A~P. Therefore in view of [14], we

retrieve sharp constants for Euclidean setting.We also like to mention that in [14], the sparse domination was obtained in

terms of norms as it was based on Lerner’s mean oscillation formula. Therefore theauthors only get the sparse domination and sharp bound for the range 1 ≤ p < ∞,but as our method depends on pointwise domination, our result is true for therange 1

2 < p < ∞.

Our strategy for the proof of Theorem 2.5 depends on two ingredients.Firstly, we need to define an appropriate “grand maximal truncated op-erator” and secondly, it is quite natural from the ideas developed in [12]and [24] that we need some sort of end-point boundedness of the “grandmaximal truncated operator” to conclude our result and to address this wehave to first address the end-point boundedness of the bilinear Calderon-Zygmund operator in this setting. The following end-point estimate is anextension of the result (page 51 ,[15]) from polynomial growth measure toupper doubling, geometric doubling metric measure.

Theorem 2.8. Let T be a bilinear Calderon-Zygmund operator, i.e the associ-ated kernel K satisfies all the properties in Definition 2.1 and T is bounded fromLp1(X, d, µ)×Lp2(X, d, µ) to Lp(X, d, µ) where 1

p= 1

p1+ 1

p2with 1 < p, p1, p2 <

∞. Then T is bounded from L1(X, d, µ) × L1(X, d, µ) to L12,∞(X, d, µ).

We also need the following version of the Cotlar’s inequality in orderto conclude the end-point boundedness of the maximally truncated bilin-ear Calderon-Zygmund operators. To state the result we need to definefollowing operators.

(7) M (f)(x) := supr>0

1

µ(B(x, 5r))

∫

B(x,r)|f(y)|dµ(y)

(8) Mλ(f)(x) := supr>0

1

λ(x, r)

∫

B(x,r)|f(y)|dµ(y)

From Vitali covering lemma it is straightforward to see both M and Mλ arebounded from L1(X, d, µ) to L1,∞(X, d, µ).

Theorem 2.9. Let T be a bilinear Calderon-Zygmund operator defined on an up-per doubling, geometrically doubling metric measure space (X, d, µ). Then T ∗ is

bounded from L1(X, d, µ)×L1(X, d, µ) to L12,∞(X, d, µ), in particular T ∗ satis-

fies the Cotlar’s type inequality:

T ∗(f1, f2)(x) ≤ Cη,λ,TMλ(f1)(x)Mλ(f2)(x) + Mη(|T (f1, f2)|)(x)

+ M (f1)(x)M (f2)(x),(9)

where η ∈ (0, 12) and Mη(f) := (M (|f |η))1η .

Presentation of the paper: The proof of Theorem 2.8 and Theorem 2.9is quite independent of the sparse domination principle, therefore to main-tain the flow of the paper we postpone them to Section 4. In Section 3,


we provide proof of Theorem 2.5 and quantitative weighted estimates areachieved in Subsection 3.2.

3. SPARSE DOMINATION

The proof of the theorem is quite involved and it uses the bilinear grandmaximal truncated operator and a recursive decomposition of David-Mattilacells in order to get the sparse domination. The notion of dyadic latticein Rn with a non-doubling measure µ was introduced by David and Mat-tila in [4]. In [24], the authors observed that the same construction of David-Mattila cells works in general in the case of a geometrically doubling metricmeasure space.

Lemma 3.1 ([4, 24]). Let (X, d, µ) be a geometrically doubling metric measurespace with doubling dimension N and locally finite Borel measure µ. If W denotesthe support of µ and C0 > 1 and A0 > 5000C0 are two given constants, then foreach integer k, there exists a partition of W into Borel sets Dk = {Q}Q∈Dk

withthe following properties

i) For each k ∈ Z, the set W is disjoint union W = ∪Q∈DkQ. Moreover, if

k < l,Q ∈ Dl, and R ∈ Dk, then either Q ∩R = ∅ or Q ⊂ R.

ii) For each k ∈ Z and each cubeQ ∈ Dk, there exists a ball B(Q) = B(zQ, r(Q))

with zQ ∈ W such that A−k0 ≤ r(Q) ≤ C0A

−k0 and W ∩ B(Q) ⊂ Q ⊂

W ∩ 28B(Q) = W ∩B(zQ, 28r(Q)). Further, the collection {5B(Q)}Q∈Dk

is pairwise disjoint.

iii) Let Ddbk denote the collection of cubes Q in Dk satisfying the doubling condi-

tion

µ(100B(Q)) ≤ C0µ(B(Q)).

iv) For the non-doubling cubes Q ∈ Dk \ Ddbk , we have r(Q) = A−k

0 and

µ(cB(Q)) ≤ C−10 µ(100cB(Q))(10)

for all 1 ≤ c ≤ C0. D will denote the collection of David-Mattila cells.

In [12], the sparse domination was achieved using a suitable stoppingtime argument with the help of the following operator

MT (f)(x) = supQ∋x

||T (fχRn\3Q)||L∞(Q).

The sparse family was then obtained with the help of the recursive useof Calderon-Zygmund decomposition. In euclidean setting the Calderon-Zygmund decomposition provides disjoint cubes and hence it is suitablefor recursive use but Tolsa’s decomposition produces “almost-disjoint” cubeswhich may not be suitable for recursive use of the decomposition as theoverlap grows very rapidly. Thus our cube selection algorithm will be dif-ferent and will rely on the structure of David-Mattila cells. In the contextof upper doubling, geometric doubling metric measure spaces the corre-sponding grand maximal truncated operator was employed in [24]. Moti-vated by that, let us define the following maximal operator for our purpose.


Definition 3.2. For a fixed cell Q0 and x ∈ Q0, the local bilinear grand maximaltruncated operator is defined by

MT,Q0(f1, f2)(x) = supP∋x,

P∈D(Q0)

‖T (f1, f2)− T (f1χ30B(P ), f2χ30B(P ))‖L∞(P ).

where D(Q0) denote the collection of David-Mattila cells contained in Q0.

In order to obtain end-point boundedness for the grand bilinear maximaltruncated operator one would like to observe the difference between MT,Q

and T ∗. The following theorem says that MT,Q and T ∗ are equivalent uptothe maximal function Mλ.

Theorem 3.3. Let Q0 be a David-Mattila cell. For functions f1, f2 supported in30B(Q0) we have the following inequality

|MT,Q0(f1, f2)(x)− T ∗(f1, f2)(x)| ≤ (CK,λ + ||ω||Dini(1))Mλ(f1, f2)(x),(11)

for all x ∈ Q0 and Mλ is the maximal function as in (5). In particular, MT,Q0

maps L1(X, d, µ) × L1(X, d, µ) to L12,∞(X, d, µ) with norm independent of Q0.

Proof. Fix x ∈ Q0 and r > 0. Observe that for large r > 0 (say r > 40r(Q0)),

Tr(f1, f2)(x) = 0, as f1, f2 are supported on 30B(Q0), so we can chose aDavid-Mattila cell P ∈ D(Q0) such that x ∈ P and r ∼ diam(P ). DenoteJk = (2k+130B(Q))2 \ (2k30B(Q))2. Then for any x′ ∈ P ,

|Tr(f1, f2)(x)| −MT,Q0(f1, f2)(x)

≤ |Tr(f1, f2)(x)| − |T (f1, f2)(x′)− T (f1χ30B(P ), f2χ30B(P ))(x

′)|

≤ |Tr(f1, f2)(x)− (T (f1, f2)(x)− T (f1χ30B(P ), f2χ30B(P ))(x))|

+ |(T (f1, f2)(x)− T (f1χ30B(P ), f2χ30B(P ))(x))− (T (f1, f2)(x′)− T (f1χ30B(P ), f2χ30B(P ))(x

′))|

≤

∫

(30B(P ))2△Sr(x)|K(x, y1, y2)||f1(y1)||f2(y2)|dµ(y1)dµ(y2)

+

∫

X2\(30B(Q))2

|K(x, y1, y2)−K(x′, y1, y2)||f1(y1)||f2(y2)|dµ(y1)dµ(y2)

≤

∫

(30B(P ))2△Sr(x)mini=1,2

1

λ(x, d(x, yi)2|f1(y1)||f2(y2)|dµ(y1)dµ(y2)

+∑

k≥0

∫

Jk

|K(x, y1, y2)−K(x′, y1, y2)|f1(y1)||f2(y2)|dµ(y1)dµ(y2)

≤ CK

∫

(30B(P ))2△Sr(x)

1

λ(x, 30r(P ))2|f1(y1)||f2(y2)|dµ(y1)dµ(y2)

+ Cλ

∑

k≥0

∫

(2k+130B(Q))2

1

λ(x, 2kr(Q))2ω

(

56r(Q)

2kr(Q)

)

|f1(y1)||f2(y2)|d~µ(y)

. (CK,λ + ||ω||Dini(1))Mλ(f1, f2)(x).


Taking supremum over all r > 0 and the fact that |Tr(f1, f2)(x)−Tr(f1, f2)(x)| .Mλ(f1, f2), we get,

T ∗(f1, f2)(x)−MT,Q0(f1, f2)(x) . (CK,λ + ||ω||Dini(1))Mλ(f1, f2)(x).

Analogously we can prove the other side to conclude

|T ∗(f1, f2)(x)−MT,Q0(f1, f2)(x)| . (CK,λ + ||ω||Dini(1))Mλ(f1, f2)(x),(12)

for x ∈ Q0. As both T ∗ (Theorem 2.9) and Mλ maps L1(X, d, µ)×L1(X, d, µ)

to L12,∞(X, d, µ) the conclusion follows automatically. �

3.1. Proof of Theorem 2.5. In this section, we provide proof of Theorem 2.5.The proof of the sparse domination (6) is constructive and follows a recur-sive argument. We shall prove a recursive formula involving the operatorMT,Q0 . Before proceeding further, we fix some notations. Let α ≥ 200 bea large number and we construct the David-Mattila cells (taking A0 suffi-

ciently large) such that 30B(Q) ⊂ 30B(Q) where Q denotes the parent ofQ. For a cell Q, we denote

A(~f ,Q) :=

2∏

i=1

1

µ(αB(Q))

∫

30B(Q)|fi|dµ

Υ(Q) :=µ(αB(Q))2

λ(zQ, αr(Q))2and M(f1, f2)(x) = sup

Q∈D

Q∋x

A(~f ,Q).

It is easy to see that Υ(Q) ≤ 1 and M is bounded from L1(X, d, µ) ×

L1(X, d, µ) to L12,∞(X, d, µ) (by Vitali covering lemma). First we provide

the following relation between the grand maximal truncated operator intwo consecutive scales of David-Mattila cells and this will be useful to pro-vide the recursive relation.

Proposition 3.4 (Consecutive scales). Let Q be any David-Mattila cell. Then

MT,Q

(f1χ30B(Q), f2χ30B(Q))(x) . CΥ(Q)A(~f , Q) +MT,Q(f1χ30B(Q), f2χ30B(Q))(x)

(13)

for all x ∈ Q.


Proof. The proof simply follows from the fact that for every cell P ∈ D(Q)with x ∈ P , we have

|T (f1χ30B(Q), f2χ30B(Q))(y)− T (f1χ30B(Q)χ30B(P ), f2χ30B(Q)χ30B(P ))(y)|

= |

∫

30B(Q)2\30B(P )2K(y, z1, z2)f1(z1)f2(z2)dµ(z1)dµ(z2)|

= |

∫

30B(Q)2\30B(P )2K(y, z1, z2)f1(z1)f2(z2)dµ(z1)dµ(z2)

+

∫

30B(Q)2\30B(Q)2K(y, z1, z2)f1(z1)f2(z2)dµ(z1)dµ(z2)|

. MT,Q(f1χ30B(Q), f2χ30B(Q))(x) +1

λ(x, r(Q))2

∫

30B(Q)2|f1(z1)||f2(z2)|dµ(z1)dµ(z2)

≤ CΥ(Q)A(~f , Q) +MT,Q(f1χ30B(Q), f2χ30B(Q))(x).

Now taking supremum over all such cells concludes the proof. �

The main argument in the proof of Theorem 2.5, depends on the follow-ing cube selection algorithm. In [12], all the cubes are doubling but in ourcase, we have to face “good” as well as “bad” cubes. In the following se-lection procedure if we face a “bad” cube we keep on subdividing it untilwe hit “good” cubes and this fact is guaranteed due to the construction ofDavid-Mattila cells. Also in this procedure, we assure the fact that the con-stants Υ(Q) decay exponentially which helps us to sum. This fact followsfrom the Lemma 3.6 in [24]. We state it for convenience.

Lemma 3.5. Let Q0 = Q1 ⊃ Q1 = Q2 ⊃ Q2 . . . and Q1, Q2, . . . are non-

doubling, then we have Υ(Qk) ≤ C−k

l02

0 Υ(Q0), where l0 is the maximal positiveinteger such that 100l0α ≤ C0.

Proof. We will only prove Υ(Q1) ≤ C−

l02

0 Υ(Q0), then the proof follows fromrecursion. Let l0 be the maximal integer such that 100l0α ≤ C0. As Q1 is abad cube, using (10), we get

µ(αB(Q1)) ≤ C−l00 µ(100l0αB(Q1)) ≤ C−l0

0 µ(αB(Q1)) = C−l00 µ(αB(Q0)),

where the last inequality requires A0 to be very large, for example A0 =CC100

0 will do. Now using the doubling property of λ we conclude, λ(zQ0 , αr(Q0)) ≤

C⌈log2 A0⌉λ λ(zQ1 , αr(Q1)). Now if we chose C0 such that C

l020 > C

⌈log2 A0⌉λ , we

get Υ(Q1) ≤ C−

l02

0 Υ(Q0) and we are done. �

Lemma 3.6. Let Q0 ∈ Ddb and f1, f2 be integrable functions with supp(fj) ⊆30B(Q0) for j = 1, 2. Then we can find a subset E ⊂ Q0, families of pairwisedisjoint David-Mattila cells Ln(Q0) ⊂ D , n = 1, 2, 3, . . . contained in E,and a family of pairwise disjoint cells F(Q0) ⊂ Ddb contained in E such thatµ(E) ≤ 1

2µ(Q0) and for every P ∈ F(Q0), Q ∈ Ln(Q0) either P ⊂ Q or


P ∩Q = ∅, and

MT,Q0(f1χ30B(Q0), f2χ30B(Q0))χQ0 ≤∑

P∈F(Q0)

MT,P (f1χ30B(P ), f2χ30B(P ))χP

+ CA(~f ,Q0) + C

∞∑

n=1

100−n∑

Q∈Ln(Q0)

A(~f,Q)χQ.(14)

The following diagram pictorially depicts our selection algorithm.

RecursiveDiagram

1.Start with adoubling cell Q0

3.Bad cells L(Q0)2.Good

cells F(Q0)

Bad child Ln(Q0)

Good child

Good Bad

Recursive use of Lemma 3.4

Final iteration of Lemma 3.4

Proof. Let Q0 ∈ Ddb, now the boundedness of the operators MT,Q0 and M

from L1(X, d, µ) × L1(X, d, µ) to L12,∞(X, d, µ) imply that for large enough

Θ > 0, we have µ(E) ≤ 12µ(Q0), where

E := {x ∈ Q0 : max(MT,Q0(f1, f2)(x),M(f1, f2)(x)) > ΘA(~f ,Q0)}.

In [12], performing Calderon-Zygmund decomposition to a set similar toE produces the required sparse family but as we have to face “good” aswell as “bad” cubes in our case our selection procedure will be differ-ent. Let L0(Q0) is the maximal David-Mattila cells inside E, this implies∑

Q∈L0(Q0)µ(Q) ≤ 1

2µ(Q0). These cells may not be doubling. For x ∈


E \⋃

L0

Q we have MT,Q0(f1, f2)(x) ≤ ΘA(~f ,Q0) and we are in a nice situa-

tion. For a cell Q ∈ L0(Q0) and x ∈ Q, the maximality of Q implies

ess supy∈Q

|T (f1, f2)(y)− T (f1χ30B(Q), f2χ30B(Q))(y)| ≤ ΘA(~f,Q0),

for all Q ( Q ⊆ Q0. Thus we get

MT,Q0(f1, f2)(x) ≤ ΘA(~f ,Q0) +MT,Q

(f1χ30B(Q), f2χ30B(Q))(x).(15)

Applying Proposition 3.4 to (15) and the fact that Υ(Q) ≤ 1, yield

MT,Q0(f1, f2)(x) ≤ KA(~f,Q0) +MT,Q(f1χ30B(Q), f2χ30B(Q))(x)(16)

for some bigger constant K . Now we keep on iterating (13) to Q (atmost afinite number of times) until we reach a “good” cube containing x and thisis guaranteed by the fact that µ(Q \

⋃

R∈S(Q)

R) = 0, where S(Q) denotes the

set of maximal good David-Mattila cells in Q (see Lemma 5.28 and 5.31 in[4]). Now the construction of David-Mattila cells ensure that this processwill terminate after a finite number of steps i.e., there can be only finitechain of nondoubling cubes Q1 ⊃ Q2 · · · ⊃ QJ such that Qj ∈ Lj andx ∈ Qj . We have already pointed out that for each such Qj , Υ(Qj) . 100−j

if we take C0 to be large enough. We iterate this procedure to each bad cellin the collection L0(Q0). Next, we separate the cells with doubling propertyand set

F(Q0) = {Q ∈ L0(Q0) : Q is doubling} and L1(Q0) = L0(Q0) \ F .

Now for a cell Q ∈ D such that Q ∈ L1(Q0), if Q is doubling we select it tothe collection F(Q0), otherwise it goes to L2(Q0). The exponential decay ofthe quantities Υ(Q) allow us to sum the intermediate levels and a recursiveapplication of this procedure concludes the proof of the lemma. �

Recall Theorem 3.3, and note that in order to prove sparse domination ofthe operator T ∗, we need sparse domination for the maximal operator Mλ.The next lemma establishes this sparse domination. Mλ is dominated bythe dyadic version of it, namely MD

λ which is defined as

MDλ (

~f)(x) := supx∈P,P∈D

Υ(P )A(~f , P ).

We require the localized version MD,Q0

λ of Mλ for doubling cell Q0, whichis defined by taking the supremum over cells contained in D(Q0), that is,

MD,Q0

λ (~f)(x) = supx∈P, P∈D(Q0)

2∏

i=1

1

λ(zP , r(P ))

∫

30B(P )|fi|dµ

= supx∈P, P∈D(Q0)

Υ(P )A(~f , P ).

Lemma 3.7. Let Q0 ∈ Ddb and f1, f2 be integrable functions with supp(fj) ⊆30B(Q0) for j = 1, 2. Then we can find a subset E ⊂ Q0, families of pairwisedisjoint cells Ln(Q0) ⊂ D , n = 1, 2, . . . contained in E, and a family of pairwise


disjoint “good” cells F(Q0) ⊂ Ddb contained in E such that µ(E) ≤ 12µ(Q0) and

for every P ∈ F(Q0), Q ∈ Ln(Q0) either P ⊂ Q or P ∩Q = ∅, and

MD,Q0

λ (~f)χQ0 ≤∑

P∈F(Q0)

MD,Pλ (~f)χP + CA(~f ,Q0)

+ C

∞∑

n=1

100−n∑

Q∈Ln(Q0)

A(~f ,Q)χQ.(17)

The proof of this lemma is similar to the previous lemma. Thus we skipthe proof. �Proof of Theorem 2.5: To obtain the sparse domination Theorem 2.5 weneed sparse domination for MT,Q0 and Mλ (see Theorem 3.3). We onlymention it for MT,Q0 .

As the upper doubling condition ensures there are plenty of “large-doubling”balls, we start with a ball D such that µ(100D) ≤ C0µ(D), also without lossof generality we may assume that the ball has radius C0. Then the construc-tion of David-Mattila cells ensures that there is a doubling-cell Q0 such thatX ′ ⊂ Q0. Let G0

0 = {Q0}. Applying Lemma 3.6 we get collection of good

cubes GQ0 as well as bad cubes LQ0n , for n = 1, 2, . . . . Define G1

n = LQ0n , for

n = 1, 2, . . . and G10 = GQ0 . Now we initiate the recursive application of

Lemma 3.6. If Gk0 is constructed, now for each doubling-cube P ∈ Gk

0 theabove-mentioned process (Lemma 3.6) produces non-doubling cubes LP

n

for n = 1, . . . and doubling cubes GP .We separate the bad cubes according to the stage they are appearing, i.e.,

define for n = 1, 2, . . . , Gk+1n =

⋃

P∈Gk0

LPn and Gk+1

0 =⋃

P∈Gk0

GP . Now one

can observe that at each step we are ensuring the following, for Q ∈ Gkn we

have∑

R∈Gk+1n ;R⊂Q

µ(R) ≤1

2µ(Q).

Gn =⋃

k

Gkn is our desired sparse families. This completes the proof of The-

orem 2.5.�

3.2. Weighted estimates.

Proposition 3.8. Let 1 < p1, p2 < ∞ be such that 1p=

2∑

i=1

1pi

. Then for any

multiple weight ~w we have

‖AS(~f)‖Lp(v~w) . Cw

2∏

i=1

‖fi‖Lpi (wi),

with the implicit constant independent of ~w and Cw as in Corollary 2.6.


Proof. We first consider the case 12 < p ≤ 1. Let us assume p1 = min{p1, p2}

and ~fσ denotes the tuple (f1σ1, f2σ2). Now,

∫

X

AS(~fσ)pv~wdµ ≤

∑

Q∈S

(

2∏

i=1

1

µ(αB(Q))

∫

30B(Q)

|fi|σidµ

)p

v~w(Q)

≤ (supQ

KQ)∑

Q∈S

µ(Q)2p(p′

1−1)

v~w(Q)p′

1−12∏

i=1

σi(αB(Q))pp′1p′i

(

2∏

i=1

∫

30B(Q)

|fi|σidµ

)p

,

where KQ =v~w(Q)p

′1

2∏

i=1σi(αB(Q))

pp′1p′i

µ(αB(Q))2pµ(Q)2p(p′1−1)

. Now using the sparseness, we observe

that

µ(Q)2p(p′

1−1) . µ(E(Q))2p(p′

1−1) ≤ v~w(Q)p′

1−12∏

i=1

σi(EQ)p(p′1−1)

p′i .

v~w(Q)p′

1−1 ≤ v~w(Q)p′

1−1 and σi(EQ)p(p′1−1)

p′i

− ppi ≤ σi(Q)

pp′1p′i−p

.

As α ≥ 200, the Vitali covering lemma ensures the boundedness of the

modified uncentered maximal function Mσf(x) := supB∋x1

σ(αB)

∫

30B |f |σdµ

from Lp(σ) to itself for 1 < p ≤ ∞, where σ(αB) denotes∫

αBσdµ. Using

this fact together with the above estimates we get,∫

X

AS(~fσ)pv~wdµ ≤ (sup

Q

KQ)∑

Q∈S

2∏

i=1

(

1

σi(αB(Q))

∫

30B(Q)

|fi|σidµ

)p

σi(EQ)ppi

≤ (supQ

KQ)2∏

i=1

{

∑

Q∈S

(

1

σi(αB(Q))

∫

30B(Q)

|fi|σidµ

)pi

σi(EQ)}

ppi

≤ (supQ

KQ)

2∏

i=1

∑

Q∈S

∫

EQ

Mσidµ(fi)piσidµ

ppi

≤ (supQ

KQ)2∏

i=1

||Mσidµ(fi)||p

Lpi (σidµ)

≤ (supQ

KQ)

2∏

i=1

‖fi‖p

Lpi(σidµ).

Hence, ‖AS(~f)‖Lp(v~w) ≤ Cw

∏2i=1 ‖fi‖Lpi (wi), whereCw = sup

Q

v~w(Q)p′1p

2∏

i=1σi(αB(Q))

p′1p′i

µ(αB(Q))2µ(Q)2(p′1−1)

.

If the underlying measure is doubling then µ(αB(Q)) and µ(B(Q)) arecomparable, we observe,

Cw .α

v~w(αB(Q))p′1p

2∏

i=1

σi(αB(Q))pp′1p′i

µ(αB(Q))2µ(αB(Q))2(p′

1−1)≃ [~w]

p′1p

A~P(µ)

Therefore, for doubling measure, we have ‖AS(~f)‖Lp(v~w) . [~w]p′

1p

A~P(µ)

∏2i=1 ‖fi‖Lpi (wi).


For p > 1, we use the duality . Let g ∈ Lp′(v~w) be a non-negative function

and ~fσ = (f1σ1, f2σ2),

∫

X

AS(~fσ)gv~wdµ

≤∑

Q∈S

∫

Q

gv~wdµ ·2∏

i=1

1

µ(αB(Q))

∫

30B(Q)

|fi|σidµ

=∑

Q∈S

v~w(Q)2∏

i=1

σi(200B(Q))

µ(αB(Q))21

v~w(Q)

∫

Q

gv~wdµ

2∏

i=1

1

σi(200B(Q))

∫

30B(Q)

|fi|σidµ

. supQ

v~w(Q)2∏

i=1

σi(200B(Q))

µ(αB(Q))2v~w(EQ)1

p′

2∏

i=1

σi(EQ)1pi

∑

Q∈S

(1

v~w(Q)

∫

Q

gv~wdµ)p′

v~w(EQ)

1

p′

2∏

i=1

∑

Q∈S

(1

σi(200B(Q))

∫

30B(Q)

|fi|σidµ)piσi(EQ)

1pi

.

Denote LQ =v~w(Q)

2∏

i=1σi(200B(Q))

µ(αB(Q))2v~w(EQ)1p′

2∏

i=1σi(EQ)

1pi

. Hence we obtain

∫

X

AS(~fσ)gv~wdµ . (supQ

LQ)‖MD

v~wdµg‖Lp′(v~wdµ)

2∏

i=1

‖Mσidµ(fi)‖Lpi (σi)

. (supQ

LQ)‖g‖Lp′(v~wdµ)

2∏

i=1

‖fi‖Lpi (σi).

Thus using duality we get ‖AS(~f)‖Lp(v~w) ≤ (supQ

LQ)∏2

i=1 ‖fi‖Lpi (wi). Now

there are two cases.

Case1: p ≥ maxi

p′i.

Now observe that,

µ(Q)2(p−1) . µ(EQ)2(p−1) . v~w(EQ)

p−1p

2∏

i=1

σi(EQ)p−1

p′i ,

and as EQ ⊂ Q, we have σi(EQ)p

p′i ≤ σi(200B(Q))

p

p′i−1

σi(EQ) for all i = 1, 2.Using these facts we get sup

Q

LQ ≤ Cw where,

Cw = supQ

v~w(Q)2∏

i=1σi(200B(Q))

p

p′i

µ(αB(Q))2µ(Q)2(p−1).


Similar to the previous case if we restrict ourselves to doubling measuresone can see that Cw ≃ [~w]A~P

. Therefore, for the doubling measure we get

‖AS(~fσ)‖Lp(v~w) . [~w]A~P(µ)

2∏

i=1

‖fi‖Lpi (σi).

Case2: Finally we consider the general case of exponents. Hence with-out loss of generality we may assume that p′1 ≥ max{p, p′2}. Observe thefollowing facts:

µ(Q)2(p′1−1) . µ(EQ)

2(p′1−1) . v~w(EQ)p′1−1

p σ1(EQ)1p1 σ2(EQ)

p′1−1

p′2

and σ2(EQ)p′1p′2−1

≤ σ2(200B(Q))p′1p′2−1

.

Using the above estimates it is easy to see that

Cw = supQ

LQ ≤ supQ

v~w(Q)p′1p σ1(200B(Q))σ2(200B(Q))

p′1p′2

µ(αB(Q))2µ(Q)2(p′1−1)

≃ [~w]p′1p

A~P,

if the underlying measure is doubling. This completes the proof of Propo-sition 3.8. �

Combining Theorem 2.5 and Proposition 3.8 yields the quantitative weightedestimates for bilinear Calderon-Zygmund operators on non-homogeneoussetting and therefore concludes the proof of Corollary 2.6.

4. END-POINT ESTIMATES

4.1. Proof of Theorem 2.8. In this subsection we prove Theorem 2.8. Asmentioned earlier this result was proved on Rn with polynomial growthmeasures in [15]. One of the difficulty in our case is to get the required es-timates after decomposing the integrals over annular regions and to over-come that we shall need some extra lemmas. The Calderon-Zygmund de-composition in this setting was given by Tolsa for polynomial growth mea-sures on Rn (see [22]) then in [1] it was extended to non-homogeneous set-ting. For ζ, τ > 1, we say a ball B is (ζ, τ) doubling if µ(ζB) ≤ τµ(B). Wewill strictly follow the Calderon-Zygmund decomposition from [1] (Lemma 6.1).We also need the following lemma which helps us to use the size conditionof the kernel appropriately.

Lemma 4.1. Let (X, d, µ) be a upper doubling, geometric doubling metric mea-sure space. Let x ∈ X, ǫ > 0. Then

∫

d(x,z)>ǫ

1

λ(x, d(x, z))2dµ(z) .

1

λ(x, ǫ).

Proof. Denote ǫ0 = ǫ. Let ǫ1 be the smallest 2kǫ such that λ(x, 2kǫ0) >2λ(x, ǫ0) where k ∈ N. ǫ2 be the smallest 2kǫ1 such that λ(x, 2kǫ1) > 2λ(x, ǫ1)holds and so on. Let ki be the smallest corresponding to ǫi, i.e, ǫi+1 = 2kiǫi


and λ(x, 2kiǫi) > 2λ(x, ǫi), also λ(x, 2ki−1ǫi) ≤ 2λ(x, ǫi). Thus collecting allthis∫

d(x,z)>ǫ

1

λ(x, d(x, z))2dµ(z) ≤

∑

i

∫

B(x,ǫi+1)\B(x,ǫi)

1

λ(x, d(x, z))2dµ(z)

.∑

i

µ(B(x, ǫi+1))

λ(x, ǫi+1)2.∑

i

1

2iλ(x, ǫ).

1

λ(x, ǫ).

If for some ǫi0 there is no k satisfying our algorithm, then λ(x, 2kǫi0) ≤2λ(x, ǫi0) for all k ≥ 1. Then µ(X) = limk→∞ µ(B(x, 2kǫi0)) ≤ 2λ(x, ǫi0) <∞. Then∫

d(x,z)>ǫ

1


≤i0−2∑

i≥0

∫

B(x,ǫi+1)\B(x,ǫi)

1

λ(x, d(x, z))2dµ(z) +

∫

X\Bi0−1

1


.

i0−2∑

i

1

2iλ(x, ǫ)+

2

λ(x, ǫ)

. Cµ1

λ(x, ǫ).

�

The following lemma is from [1]. One needs some minor correction inthe proof given in [1].

Lemma 4.2. Let (X, d, µ) be a upper doubling, geometric doubling space. LetQ ⊂ R be concentric balls such that there are no (ζ, τ)- doubling balls (with

τ > Clog2 ζλ ) of the form ζkQ, k ≥ 0, with Q ⊂ ζkQ ⊂ R, then

∫

R\Q

1

λ(x, d(x, cQ))dµ(x) ≤ C,

where C depends only on ζ, τ, and µ.

The decomposition and idea of the proof are based on Proposition 5.2 in[15] except for certain estimates. For convenience, we are including a de-tailed sketch of the proof.

Before going into the proof let us collect some important ingredientswhich will be helpful. Let f1, f2 ∈ L1(µ) with ‖fi‖L1(X,d,µ) = 1 with i = 1, 2and t > 0.

We perform Calderon-Zygmund decomposition (Lemma 6.1 in [1]) to

f1 at height t12 and get disjoint balls {Q1,i} and R1,i is the smallest (3 ×

62, Clog2 3×62+1λ )-doubling ball of the form {(3 × 62)kQ1,i}k≥1 satisfying the

following properties. We write f1 = g1 + β1 = g1 +∑

i β1,i, where,

i) 1µ(62Q1,i)

∫

Q1,i|f1|dµ & t

12 and |f1(x)| ≤ t

12 for all x ∈ X \ ∪i6Q1,i.

ii) 1µ(62ηQ1,i)

∫

ηQ1,i|f1|dµ . t

12 for all η > 1.


iii) g1 = f1χX\∪6Q1,i+∑

i

φ1,i, β1,i = ω1,if1 − φ1,i with β1,i(Q1,i) = 0 and

‖β1,i‖L1(X,d,µ) . |f1|(Q1,i).

iv) ‖g1‖L∞(X,d,µ) . t12 , ‖g1‖L1(X,d,µ) ≤ 1 and ‖g1‖

sLs(X,d,µ) . t

s−12 for 1 <

s < ∞.

v) supp(φ1,i) ⊂ R1,i, ‖φ1,i‖L∞(X,d,µ)µ(R1,i) . |f1|(Q1,i).

Apply Calderon-Zygmund decomposition to f2 also at the same height (t12 )

and corresponding notations will be g2, Q2,j, R2,j , β2,j , φ2,j , ω2,j etc. As a byproduct of Calderon-Zygmund decomposition, we also have the followinglemma.

Lemma 4.3. Let fi be as above and {Qi,j}j are cubes associated to the Calderon-

Zygmund decomposition of fi at height t12 for i = 1, 2. Then for i = 1, 2,

supr>0

|fi|(B(x, r))

λ(x, r)≤ κ t

12 ,

for all x ∈ X \⋃

j 62Qi,j . The constant κ only depends on Cλ and the doubling

dimension of X.

Proof. We only prove it for i = 2. If B(x, r)⋂

(⋃

6Q2,j) = ∅. Then by

Calderon-Zygmund decomposition (as |f2| ≤ t12 on (

⋃

6Q2,j)c) we get,

|f2|(B(x, r)) ≤ t12µ(B(x, r)) ≤ Cλt

12λ(x, r).

If B(x, r)∩6Q2,j 6= ∅, then as x ∈ X \⋃

62Q2,j , we have r > 28r(Q2,j). ThusB(x, r) ⊂ B(cQ2,j , Cr) and this implies together with Calderon-Zygmund de-composition

|f2|(B(x, r)) ≤ |f2|(B(cQ2,j , Cr)) . t12µ(B(cQ2,j , 6

2Cr)) ≤ κt12λ(x, r),

where we have used λ(x,Cr) ≃ λ(cQ2,j , Cr) as x ∈ B(cQ2,j , Cr). �

Let us start with the proof of Theorem 2.8.

Proof of Theorem 2.8:

Proof. We will prove that

µ({x ∈ X : |Tǫ(f1, f2)(x)| > t}) .1

t12

(18)

with constant independent of ǫ > 0.

µ({x ∈ X : |Tǫ(f1, f2)(x)| > t})

≤ µ(⋃

i

62Q1,i) + µ(⋃

j

62Q2,j) + µ(IGG) + µ(IBG) + µ(IGB) + µ(IBB),


where

IGG := {x ∈ X : |Tǫ(g1, g2)(x)| >t

4},

IBG := {x ∈ X \⋃

i

62Q1,i : |Tǫ(β1, g2)(x)| >t

4},

IGB := {x ∈ X \⋃

j

62Q2,j : |Tǫ(g1, β2)(x)| >t

4},

IBB := {x ∈ X \ (⋃

i

62Q1,i ∪⋃

j

62Q2,j) : |Tǫ(β1, β2)(x)| >t

4}.

By the decomposition

µ(⋃

i

62Q1,i) .1

t12

∑

i

∫

Q1,i

|f1| .1

t12

.

Similarly µ(⋃

j 62Q2,j) .

1

t12

.

Good-Good: Using the boundedness of T we get

µ(IGG) .1

tp‖g1‖

pLp1 (X,d,µ)‖g2‖

pLp2 (X,d,µ) .

1

tptp2(p1−1p1

+p2−1p2

).

1

tptp−

12 .

1

t12

.

(19)

Bad-Good: We want to estimate IBG := {x ∈ X\⋃

i

62Q1,i : |Tǫ(β1, g2)(x)| >

t4}. Observe

µ(IBG) .1

t

∑

i

∫

X\2R1,i

|Tǫ(β1,i, g2)(x)|dµ(x) +1

t

∑

i

∫

2R1,i\62Q1,i

|Tǫ(β1,i, g2)(x)|dµ(x)

Consider the first term. Let x /∈ (2R1,i) be such that dist(x,R1,i) > ǫ, thensupp(β1,i) ⊂ R1,i. Therefore using the cancellation in the first componentwe get

|Tǫ(β1,i, g2)(x)|

= |

∫

d(x,z)>ǫ

∫

R1,i

(K(x, y, z) −K(x, y′, z))β1,i(y)dµ(y)g2(z)dµ(z)|

.

∫

d(x,z)>ǫ

∫

R1,i

ω

(

d(y, cQ1,i)

(d(x, y) + d(x, z))

)

|β1,i(y)|dµ(y)||g2|dµ(z)

(λ(x, d(x, y)) + λ(x, d(x, z)))2

. t12 ||β1,i||1ω(

r(R1,i)

d(x, cR1,i ))

∫

d(x,z)>ǫ

1

(λ(x, d(x,R1,i)) + λ(x, d(x, z)))2dµ(z)

. t12 ||β1,i||1ω(

r(R1,i)

d(x, cR1,i ))

(

∫

ǫ<d(x,z)≤d(x,R1,i)

1

λ(x, d(x,R1,i))2dµ(z)

)

+ t12 ||β1,i||1ω(

r(R1,i)

d(x, cR1,i ))

(

∫

d(x,z)>d(x,R1,i)

1


)

. t12 ||β1,i||1ω(

r(R1,i)

d(x, cR1,i ))

1

λ(x, d(x,R1,i))(by Lemma 4.1.)


For the first term we use the doubling condition on R1,i and boundednessof Tǫ to conclude the following:

Sa . µ(R1,i)1− 1

p ‖φ1,i‖L∞(X,d,µ)µ(R1,i)1p1 t

12µ(R1,i)

1p2

. t12 |f1|(Q1,i)(25)

For Sb, use Lemma 4.1 and observe

|Tǫ(φ1,i, χ(4R1,i)cg2)(x)| ≤

∫

(4R1,i)c

∫

R1,i

|φ1,i(y)|dµ(y)|g2(z)|dµ(z)

(λ(x, d(x, y)) + λ(x, d(x, z)))2

. ‖φ1,i‖L∞(X,d,µ)t12µ(R1,i)

1

λ(x,Cr(R1,i)).

Now

Sb . ‖φ1,i‖L∞(X,d,µ)t12µ(R1,i)

∫

2R1,i

1

λ(x,Cr(R1,i))dµ(x)

. t12 |f1|(Q1,i).(26)

Thus from (25) and (26), we get T2 . t12 |f1|(Q1,i). Combining this with (23)

and (24), we get

µ(IBG) .1

tt12

∑

i

|f1|(Q1,i) .1

t12

.(27)

The estimate for IGB follows exactly from the arguments as in IBG. Thuswe skip it.Bad-Bad: BB := {x ∈ X \ B : |Tǫ(β1, β2)(x)| >

t4} where B =

⋃

i

62Q1,i ∪⋃

j

62Q2,j .

Tǫ(β1, β2) =∑

i

Tǫ

β1,i,∑

j:r(R1,i)≤r(R2,j)

β2,j

+∑

j

Tǫ

∑

i:r(R1,i)>r(R2,j))

β1,i, β2,j

.

The above two terms are symmetric and thus we only treat the first. DenoteTi := {j : r(R1,i) ≤ r(R2,j)}.

µ({x ∈ X \ B :∑

i

|Tǫ(β1,i,∑

j∈Ti

β2,j)(x)| >t

8})

≤ µ({x ∈ X \ B :∑

i

χ(2R1,i)c |Tǫ(β1,i,∑

j∈Ti

β2,j)(x)| >t

16})

+ µ({x ∈ X \ B :∑

i

χ2R1,i |Tǫ(β1,i,∑

j∈Ti

β2,j)(x)| >t

16})

=: S1 + S2.(28)

We only give the estimate for S1. Estimates for S2 follows from a similarline of arguments and incorporating the ideas from [3, 15].

Estimate for S1:

S1 . Ia1 + Ia2 + Ia3 + Ia4 + Ib,


where,

Ia1 :=1

t

∫

X\B

∑

i,j

χ(2R1,i)cχ(2R2,j )cχmin{d(x,R1,i),d(x,R2,j)}>ǫ|Tǫ(β1,i, β2,j)(x)|

dµ(x),

Ia2 :=1

t12

∫

X\B

∑

i,j

χ(2R1,i)cχ(2R2,j )cχ{d(x,R1,i)>ǫ, d(x,R2,j)≤ǫ}|Tǫ(β1,i, β2,j)(x)|

12

dµ(x),

Ia3 :=1

t12

∫

X\B

∑

i,j

χ(2R1,i)cχ(2R2,j )cχ{d(x,R2,j )>ǫ, d(x,R1,i)≤ǫ}|Tǫ(β1,i, β2,j)(x)|

12

dµ(x),

Ia4 :=1

t12

∫

X\B

∑

i,j

χ(2R1,i)cχ(2R2,j )cχ{max{d(x,R1,i),d(x,R2,j)}≤ǫ}|Tǫ(β1,i, β2,j)(x)|

12

dµ(x),

Ib :=1

t12

∫

X\B

∑

i,j

χ(2R1,i)cχ(2R2,j )|Tǫ(β1,i, β2,j)(x)|

12

dµ(x).

Estimate for Ia1:Using cancellation at the first component (for β1,i), |Tǫ(β1,i, β2,j)(x)| is dom-inated by

||β1,i||1||β2,j ||1ω

(

r(R1,i)

(d(x, cR1,i) + d(x, cR2,j ))

)

1

(λ(x, d(x, cR1,i )) + λ(x, d(x, cR2,j )))2

.||β1,i||1

∫

χQ2,jω

(

r(R1,i)

(d(x, cR1,i ) + d(x, cR2,j ))

)

|f2(z)|dµ(z)

(λ(x, d(x, cR1,i )) + λ(x, d(x, z)))2.

Now we sum over all j and as∑

j χQ2,j ≤ C , the sum is dominated by thefollowing

||β1,i||1

∫

d(x,z)>ǫ

ω

(

r(R1,i)

(d(x, cR1,i ) + d(x, cR2,j ))

)

|f2(z)|dµ(z)

(λ(x, d(x, cR1,i )) + λ(x, d(x, z)))2

Now we split the last quantity as following:

||β1,i||1

∫

ǫ<d(x,z)≤d(x,cR1,i)ω

(

r(R1,i)

(d(x, cR1,i ) + d(x, cR2,j ))

)

|f2(z)|dµ(z)

(λ(x, d(x, cR1,i )) + λ(x, d(x, z)))2

+ ||β1,i||1

∫

d(x,z)>d(x,cR1,i)ω

(

r(R1,i)

(d(x, cR1,i) + d(x, cR2,j ))

)

|f2(z)|dµ(z)

(λ(x, d(x, cR1,i )) + λ(x, d(x, z)))2.

Now using Lemma 4.3, the first term is simply dominated by

κ||β1,i||1ω

(

r(R1,i)

d(x, cR1,i)

)

t12

λ(x, d(x, cR1,i )).

Now for the second term construct a sequence of radii as in the Lemma 4.1and call them {rl}. Denote r0 = d(x, cR1,i) and let kl be the smallest positive

integer corresponding to rl, i.e, rl+1 = 2klrl such that λ(x, 2klrl) > 2λ(x, rl),


and λ(x, 2kl−1rl) ≤ 2λ(x, rl), then the integral is dominated by

||β1,i||1ω(r(R1,i)

d(x, cR1,i))∑

l

∫

B(x,rl+1)\B(x,rl)

|f2(z)|


≤ ||β1,i||1ω(r(R1,i)

d(x, cR1,i ))∑

l

|f2|(B(x, rl+1))

λ(x, rl+1)2

. κ||β1,i||1ω(r(R1,i)

d(x, cR1,i ))∑

l

t12λ(x, rl+1)

λ(x, rl+1)2(by Lemma 4.3)

. ||β1,i||1ω(r(R1,i)

d(x, cR1,i ))

t12

λ(x, r0)

∑

l

1

2l. ||β1,i||1ω

(

r(R1,i)

d(x, cR1,i)

)

t12

λ(x, d(x, cR1,i )).

Now first integrating over X \ 2R1,i and then summing over i, gives that

(29) Ia1 .‖ω‖Dini(1)

t12

.

Estimate for Ia2:For d(., R1,i) > ǫ, R1,i ⊂ B(x, ǫ)c, we can use the cancellation in the firstcomponent to obtain the following:

|Tǫ(β1,i, β2,j)(x)| . ω

(

r(R1,i)

d(x, cR1,i)

)

|f1|(Q1,i)

λ(x, d(x, cR1,i ))

|f2|(Q2,j)

λ(x, ǫ)

As d(x,R2,j) ≤ ǫ, we have d(x, cR2,j ) ≤ Cǫ. Thus using the above estimatewith Holder’s inequality and subsequently using the annular decomposi-tion we get

Ia2 .1

t12

(

∑

i

∑

k

∫

2k+1r(R1,i)≥d(x,cR1,i)>2kr(R1,i)

ω

(

r(R1,i)

d(x, cR1,i)

)

|f1|(Q1,i)

λ(x, d(x, cR1,i ))

) 12

∑

j

∫

B(cR2,j,Cǫ)

|f2|(Q2,j)

λ(x, ǫ)dµ(x)

12

.‖ω‖

12

Dini(1)

t12

.

(30)

Estimate for Ia3: Similar to Ia2.

Estimate for Ia4:When x ∈ (2R1,i)

c ∩ (2R2,j)c with d(x,R1,i) ≤ ǫ, d(x,R2,j) ≤ ǫ. We have,

|Tǫ(β1,i, β2,j)(x)| .|f1|(Q1,i)

λ(x, ǫ)

|f2|(Q2,j)

λ(x, ǫ).


This implies

Ia4 .1

t12

(

∑

i

∫

B(cR1,i,Cǫ)

|f1|(Q1,i)

λ(x, ǫ)dµ(x)

) 12

∑

j

∫

B(cR2,j,Cǫ)

|f2|(Q2,j)

λ(x, ǫ)dµ(x)

12

.1

t12

(

∑

i

|f1|(Q1,i)

) 12

∑

j

|f2|(Q2,j)

12

.1

t12

.

(31)

Estimate for Ib:

Ib ≤1

t12

∫

X\B

∑

i,j

χ(2R1,i)cχ(2R2,j )|Tǫ(β1,i, φ2,j)(x)|

12

dµ(x)

+1

t12

∫

X\B

∑

i,j

χ(2R1,i)cχ(2R2,j )\62Q2,j|Tǫ(β1,i, w2,jf2)(x)|

12

dµ(x)

:= I ′b + I ′′b .(32)

Estimate for I ′b:Case 1: For I ′b, consider x ∈ (2R1,i)

c ∩ 2R2,j such that d(x,R1,i) > ǫ. Thenusing the cancellation in the first component, similar to (20), we get

|Tǫ(β1,i, φ2,j)(x))| . |f1|(Q1,i)‖φ2,j‖L∞(X,d,µ)1

λ(x, d(x,R1,i))ω

(

r(R1,i)

d(x, cR1,i)

)

.

Now using Holder’s inequality and the above estimate we get,

1

t12

∫

X\B

∑

i,j

χ(2R1,i)cχd(.,R1,i)>ǫχ(2R2,j )|Tǫ(β1,i, φ2,j)(x)|

12

dµ(x)

.1

t12

∑

i

|f1|(Q1,i)∑

k

∫

2kr(R1,i)≤d(x,cR1,i)<2k+1r(R1,i)

1

λ(x, d(x,R1,i))ω

(

r(R1,i)

d(x, cR1,i )

)

12

×

∑

j

‖φ2,j‖L∞(X,d,µ)µ(R2,j)

12

.

(

∑

i

|f1|(Q1,i)∑

k

ω(2−k)

) 12

∑

j

|f2|(Q2,j)

12

.‖ω‖

12

Dini(1)

t12

.

(33)


Case 2: Consider x ∈ (2R1,i)c ∩ 2R2,j with d(x,R1,i) ≤ ǫ. Then the size

condition with Lemma 4.1 gives the following estimate:

|Tǫ(β1,i, φ2,j)(x)|

. |f1|(Q1,i)‖φ2,j‖L∞(X,d,µ)

(

∫

d(x,z)>ǫ

dµ(z)

λ(x, d(x, z))2+

∫

d(x,z)≤ǫ,d(x,y)>ǫ

1

λ(x, ǫ)2dµ(z)

)

. |f1|(Q1,i)‖φ2,j‖L∞(X,d,µ)1

λ(x, ǫ).

Now using the above estimate we get

1

t12

∫

X\B

∑

i,j

χ(2R1,i)cχ(2R2,j)χd(.,R1,i)≤ǫ|Tǫ(β1,i, φ2,j)(x)|

12

dµ(x)

.1

t12

(

∫

X

∑

i

χB(cR1,i,Cǫ)

|f1|(Q1,i)

λ(x, ǫ)

) 12

∫

X

∑

j

χ2R2,j‖φ2,j‖L∞(X,d,µ)

12

.1

t12

(∑

i

|f1|(Q1,i))12 (∑

j

|f2|(Q2,j))12 .

1

t12

.

(34)

Combining (33),(34) we get

I ′b .1

t12

.(35)

Estimate for I ′′b :

Case 1: Consider x such that d(x,R1,i) > ǫ. Then using cancellation at thefirst component we get

|Tǫ(β1,i, w2,jf2)(x)| . ω

(

r(R1,i)

d(x, cR1,i)

)

|f1|(Q1,i)

λ(x, d(x, cR1,i ))

|f2|(Q2,j)

λ(x, d(x, cR2,j )).


Now using Lemma 4.2 at the last but one step we get,

1

t12

∫

X\B

∑

i,j

χ(2R1,i)cχ(2R2,j )\62Q2,jχd(.,R1,i)>ǫ|Tǫ(β1,i, w2,jf2)(x)|

12

dµ(x)

.1

t12

(

∫

X

∑

i

χ(2R1,i)cω

(

r(R1,i)

d(x, cR1,i )

)

|f1|(Q1,i)

λ(x, d(x, cR1,i ))dµ(x)

) 12

×

∫

X

∑

j

χ(2R2,j )\62Q2,j

|f2|(Q2,j)

λ(x, d(x, cR2,j ))dµ(x)

12

.‖ω‖

12

Dini(1)

t12

(∑

i

|f1|(Q1,i))12

∑

j

∫

2R2,j\62Q2,j

|f2|(Q2,j)

λ(x, d(x, cR2,j ))dµ(x)

12

.‖ω‖

12

Dini(1)

t12

.

(36)

Case 2: For x ∈ (2R1,i)c ∩ (62Q2,j)

c with d(x,R1,i) ≤ ǫ. Then

|Tǫ(β1,i, w2,jf2)(x)| .

∫

R2,j

∫

R1,i

1

(λ(x, d(x, y)) + λ(x, d(x, z)))2|β1,i(y)||w2,jf2|dµ(y)dµ(z)

.|f1|(Q1,i)

λ(x, ǫ)

|f2|(Q2,j)

λ(x, d(x, cR2,j )).

Now using the above estimate and Lemma 4.2,

1

t12

∫

X\B

∑

i,j

χ(2R1,i)cχ(2R2,j)\62Q2,jχd(.,R1,i)≤ǫ|Tǫ(β1,i, w2,jf2)(x)|

12

dµ(x)

.1

t12

(

∑

i

|f1|(Q1,i)

∫

B(cR1,i,Cǫ)

dµ(x)

λ(x, ǫ)

)12

∑

j

|f2|(Q2,j)

∫

(2R2,j)\62Q2,j

dµ(x)

λ(x, d(x, cR2,j ))

12

.1

t12

.

(37)

This completes the estimate for I ′′b . We get from (36) and (37)

I ′′b .1

t12

.(38)

Thus

Ib .1

t12

.(39)

From (29), (30), (31), (39) we get

S1 .CT,‖ω‖Dini(1)

t12

.(40)


Arguing along the ideas developed in [3, 15] and providing necessary mod-ifications (as we have given for S1) we can prove

S2 .CT,‖ω‖Dini(1)

t12

.(41)

From (28), (40) and (41) we get

µ({x ∈ X \ B :∑

i

|Tǫ(β1,i,∑

j∈Ti

β2,j)(x)| >t

8}) .

CT,‖ω‖Dini(1)

t12

.(42)

These estimates imply µ(IBB) .CT,‖ω‖Dini(1)

t12

. This completes the proof. �

4.2. Cotlar’s lemma. In this subsection, end-point boundedness for themultilinear maximal Calderon-Zygmund operator will be addressed. Wewill conclude this with a Cotlar’s inequality for multi-linear Calderon-Zygmundoperators on non-homogeneous spaces. Cotlar’s inequality for Calderon-Zygmund operators for polynomially growth measures was first obtainedin [16]. The proof substantially depends on the Guy-David lemma men-tioned in [16]. In [9], the authors extended this to the general framework ofnon-homogeneous spaces. The proof the Theorem 2.9 requires the follow-ing lemma which enables us to pass from arbitrary balls to “doubling ball”.The idea of the proof is inspired by Lemma 3.1 in [9].

Lemma 4.4. Let T be a bilinear Calderon-Zygmund operator defined on an upperdoubling, geometrically doubling metric measure space (X, d, µ). Then for anyx ∈ supp(µ), r > 0, there is an integer k = k(x, r) ∈ N such that

|Tr(f1, f2)(x)− T5R(f1, f2)(x)| .λ,µ M f1(x)M f2(x)(43)

where R = 5k−1r and the ball B(x,R) is doubling i.e., µ(B(x, 25R)) . µ(B(x,R)).

Proof. Let x ∈ supp(µ) and r > 0. We claim that there is some j ∈ N suchthat µ(B(x, 5j+1r)) ≤ 4C6

λµ(B(x, 5j−1r)). If the claim is not true then usingthe upper doubling condition we get

µ(B(x, r)) ≤ (4C6λ)

−jµ(B(x, 52jr)) . (4C6λ)

−jλ(x, 52jr) ≤ 5−jλ(x, r).

Letting j → ∞ we get µ(B, r) = 0 but this is a contradiction as µ(B, r) > 0for x ∈ supp(µ). Thus the claim is true.

Let k be the smallest integer such that the claim happens thus µ(B(x, 25R)) .µ(B(x,R)) where R = rk−1 = 5k−1r. Denote µj = µ(B(x, 5jr)) for j =

1, . . . , k., then we have µj+1 ≤ (2C3λ)

j+2−kµk and λ(x, rk) ≤ (C3λ)

k−j−1λ(x, rj+1) for j =


1, . . . , k. Now using the above estimates we obtain

|Tr(f1, f2)(x) − T5R(f1, f2)(x)|

≤k−1∑

j=1

∫

B(x,rj+1)\B(x,rj)

|f1(y1)|

λ(x, d(x, y1))dµ(y1)

k−1∑

l=1

∫

B(x,rl+1)\B(x,rl)

|f2(y2)|


+1

λ(x, r)

∫

B(x,r)

|f1(y1)|dµ(y1)k−1∑

l=1

∫

B(x,rl+1)\B(x,rl)

|f2(y2)|


+

k−1∑

j=1

∫

B(x,rj+1)\B(x,rj)

|f1(y1)|


1

λ(x, r)

∫

B(x,r)

|f2(y2)|dµ(y2)

.

k−1∑

j=1

2j−kM f1(x)

k−1∑

l=1

2l−kM f2(x) + M f1(x)

k−1∑

l=1

2l−kM f2(x)

+ M f2(x)

k−1∑

j=1

2j−kM f1(x)

. M f1(x)M f2(x).

(44)

�

Proof of Theorem 2.9:For multilinear Calderon-Zygmund operators L. Grafakos and R. Torresobtained the Cotlar’s inequality in [6]. We combine their ideas with Lemma 4.4to obtain Theorem 2.9. Let x ∈ X and r > 0, then by Lemma 4.4

|Tr(f1, f2)(x)− T5R(f1, f2)(x)| .λ,µ M f1(x)M f2(x)(45)

where R = 5k−1r and µ(B(x, 25R)) . µ(B(x,R)). For any z ∈ B(x,R)

(46) T5R(f1, f2)(z) = T (f1, f2)(z) − T (f01 , f

02 )(z)

where f0i (x) = fiχB(x,5R) for i = 1, 2. DenoteSr(x) = {~y ∈ X2 : max

i=1,2d(x, yi) ≤

r} and by the regularity condition we get

|T5R(f1, f2)(x) − T5R(f1, f2)(z)|

.

∫

S5R(x)c

(

mini=1,2

1

λ(x, d(x, yi))2

)

ω(d(x, z)

d(x, y1) + d(x, y2))

2∏

i=1

|fi(yi)|d~µ

. I + II + III

where

I =

∫

d(x,y1)≤5R

∫

d(x,y2)>5R

(

mini=1,2

1

λ(x, d(x, yi))2

)

ω(d(x, z)

d(x, y1) + d(x, y2))

2∏

i=1

|fi(yi)|d~µ

≤ Mλ(f1)(x)∑

l≥0

ω(2−l)1

λ(x, 5R2l+1)

∫

d(x,y2)≤5R2l+1

|f2(y2)|dµ(y2)

. Mλ(f1)(x)Mλ(f2)(x)

and

II =

∫

d(x,y2)≤5R

∫

d(x,y1)>5R

(

mini=1,2

1

λ(x, d(x, yi))2

)

ω(d(x, z)

d(x, y1) + d(x, y2))

2∏

i=1

|fi(yi)|d~µ


Arguing as in I we obtain II . Mλ(f1)(x)Mλ(f2)(x). For III we observe

III

=

∫

d(x,y1)>5R

∫

d(x,y2)>5R

(

mini=1,2

1

λ(x, d(x, yi))2

)

ω(d(x, z)

d(x, y1) + d(x, y2))

2∏

i=1

|fi(yi)|d~µ

≤∑

j≥0

∫

d(x,y2)≤2j+15R

|f2(y1)|dµ(y2)1

λ(x, 2j+15R)2

∫

d(x,y1)≤2j+15R

|f1(y1)|dµ(y1)ω(2−j)

+∑

l≥0

l∑

j≥1

∫

2j5R<d(x,y1)≤2j+15R

|f1(y1)|dµ(y1)1

λ(x, 2l+15R)2

∫

d(x,y2)≤2l+15R

|f2|dµ(y2)ω(2−l)

. Mλ(f1)(x)Mλ(f2)(x).

Hence

(47) |T5R(f1, f2)(x)− T5R(f1, f2)(z)| . Mλ(f1)(x)Mλ(f2)(x).

Now from (46) and (47) we obtain

(48) |T5R(f1, f2)(x)| . Mλ(f1)(x)Mλ(f2)(x)+|T (f1, f2)(z)|+|T (f01 , f

02 )(z)|

for all z ∈ B(x,R). Fix 0 < η < 12 . Raising (48) to the power η and averaging

over B = B(x,R) we get

|T5R(f1, f2)(x)|η .Mλ(f1)(x)Mλ(f2)(x) + M (|T (f1, f2)|

η)(x)

+1

µ(B(x,R))

∫

B(x,R)

|T (f01 , f

02 )(z)|

ηdµ(z).(49)

Now arguing as in [6] and using the facts that T maps L1(X, d, µ)× L1(X, d, µ) to

L12 ,∞(X, d, µ) (Theorem 2.8) and µ(B(x, 25R)) . µ(B(x,R)) we obtain

1

µ(B(x,R))

∫

B(x,R)

|T (f01 , f

02 )(z)|

ηdµ(z) ≤ Cη,Tµ(B(x,R))−2η(

2∏

i=1

‖f0i ‖L1(X,d,µ))

η

≤ Cη,T

(

2∏

i=1

1

µ(B(x, 25R))

∫

B(x,5R)

|fi|dµ(yi)

)η

≤ Cη,T (M (f1)(x)M (f2)(x))η(50)

Now combining the estimates (45), (48), (49) and (50), we obtain

|Tr(f1, f2)(x)| ≤ Cη,λ,TMλ(f1)(x)Mλ(f2)(x) + Mη(|T (f1, f2)|)(x)

+ M (f1)(x)M (f2)(x).(51)

This together with Remark 2.3 proves (9) . Now using the facts that M

maps Lp,∞(X, d, µ) to Lp,∞(X, d, µ) for all 1 < p < ∞ and T maps L1(X, d, µ)×

L1(X, d, µ) to L12,∞(X, d, µ), we get T ∗ maps L1(X, d, µ) × L1(X, d, µ) to

L12,∞(X, d, µ). �

REFERENCES

[1] Bui,T. A., Duong, X. T.:Hardy spaces, regularized BMO spaces and the boundedness ofCalderon-Zygmund operators on non-homogeneous spaces, J. Geom. Anal. 23(2), 895-932(2013).

[2] Conde-Alonso, J., Rey, G.:A pointwise estimate for positive dyadic shifts and some applica-tions. Math. Ann. 365 (3-4), 1111-1135 (2016).


[3] Damian, W., Hormozi, M., Li, K.:New bounds for bilinear Caldern-Zygmund operators andapplications. Rev. Mat. Iberoam. 34(3), 1177-1210 (2018).

[4] David, G., Mattila, P.: Removable sets for Lipschitz harmonic functions in the plane, Rev.Mat. Iberoamericana. 16(1), 137-215 (2000).

[5] Grafakos, L.,Torres, R. H.: Multilinear Calderon-Zygmund theory, Adv. Math. 165(1), 124-164 (2002).

[6] Grafakos, L.,Torres, R. H.: Maximal operator and weighted norm inequalities for multilinearsingular integrals, Indiana Univ. Math. J. 51(5), 1261-1276 (2002).

[7] Hu, G., Meng, Y., Yang, D.: Weighted norm inequalities for multilinear Calderon-Zygmundoperators on non-homogeneous metric measure spaces, Forum Math. 26(5), 1289-1322 (2014).

[8] Hytonen, T. P.: A framework for non-homogeneous analysis on metric spaces, and the RBMOspace of Tolsa, Publ. Math. 54(2), 485-504 (2010).

[9] Hytonen, T. P., Liu, S., Yang, D., Yang, D.: Boundedness of Calderon-Zygmund operatorson non-homogeneous spaces, Canad. J. Math. 64 (4), 892-923 (2012).

[10] Lacey, M. T.: An elementary proof of the A2 bound, Israel J. Math. 217 (1), 181-195 (2017).[11] Lerner, A. K., Ombrosi, S., Perez, C., Torres, R. H., Trujillo-Gonzalez, R.: New maximal

functions and multiple weights for the multilinear Calderon-Zygmund theory, Adv. Math.220(4), 1222-1264 (2009).

[12] Lerner, A. K.: On pointwise estimates involving sparse operators, New York J. Math. 22,341-349 (2016).

[13] Lerner, A. K., Nazarov, F.:Intuitive dyadic calculus: the basics, Expo. Math. 37(3), 225-265(2019).

[14] Li, K., Moen, K., Sun, W.:The sharp weighted bound for multilinear maximal functions andCalderon-Zygmund operators, J. Fourier Anal. Appl. 20(4), 751-765 (2014).

[15] Martikainen, H.,Vuorinen, E.: Dyadic-probabilistic methods in bilinear analysis, arXivpreprint arXiv:1609.01706.

[16] Nazarov, F., Treil, S., Volberg, A.: Cauchy integral and Calderon-Zygmund operators onnonhomogeneous spaces, Internat. Math. Res. Notices. 15, 703-726 (1997).

[17] Nazarov, F., Treil, S., Volberg, A.:Weak type estimates and Cotlar inequalities for Calderon-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices. 9, 463-487(1998).

[18] Nazarov, F., Treil, S., Volberg, A.: The Tb-theorem on non-homogeneous spaces, Acta Math.190(2), 151-239 (2003).

[19] Tolsa, X.: Cotlar’s inequality without the doubling condition and existence of principal valuesfor the Cauchy integral of measures, J. Reine Angew. Math. 502 , 199-235 (1998).

[20] Tolsa, X.: L2-boundedness of the Cauchy integral operator for continuous measures, DukeMath. J. 98(2), 269-304 (1999).

[21] Tolsa, X.: A proof of the weak (1, 1) inequality for singular integrals with non doubling mea-sures based on a Calderon-Zygmund decomposition. Publ. Mat. 45(1), 163174 (2001).

[22] Tolsa, X.: BMO, H1, and Calderon-Zygmund operators for non doubling measures, Math.Ann. 319(1), 89-149 (2001).

[23] Tolsa, X.:Painleve’s problem and the semiadditivity of analytic capacity, Acta Math. 190 (1),105-149 (2003).

[24] Volberg, A., Zorin-Kranich, P.: Sparse domination on non-homogeneous spaces with an ap-plication to Ap weights, Rev. Mat. Iberoamericana. 34(3), 1401-1414 (2018).

ABHISHEK GHOSH, DEPARTMENT OF MATHEMATICS AND STATISTICS, INDIAN INSTI-TUTE OF TECHNOLOGY KANPUR, KANPUR-208016, INDIA.

E-mail address: [email protected]

ANKIT BHOJAK, DEPARTMENT OF MATHEMATICS AND STATISTICS, INDIAN INSTITUTE

OF TECHNOLOGY KANPUR, KANPUR-208016, INDIA.E-mail address: [email protected]

PARASAR MOHANTY, DEPARTMENT OF MATHEMATICS AND STATISTICS, INDIAN IN-STITUTE OF TECHNOLOGY KANPUR, KANPUR-208016, INDIA.


arXiv:1609.01706


SAURABH SHRIVASTAVA, DEPARTMENT OF MATHEMATICS, INDIAN INSTITUTE OF SCI-ENCE EDUCATION AND RESEARCH BHOPAL, BHOPAL-462066, INDIA.


arxiv:1609.08995v2 [math.ca] 5 mar 2019arxiv:1609.08995v2 [math.ca] 5 mar 2019 sharp weighted...

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