Boundedness for Commutators of Bilinear [theta] -Type Calderón-Zygmund Operators on Nonhomogeneous Metric Measure Spaces

Abstract

Let ( X , d , μ ) be a nonhomogeneous metric measure space. In this paper, the boundedness for commutators generated by bilinear θ -type Calderón-Zygmund operators and R B M O ( μ ) functions on ( X , d , μ ) is obtained.

[ProQuest: [...] denotes non US-ASCII text; see PDF]

Rulong Xie 1,2 and Lisheng Shu 3 and Aiwen Sun 3

Academic Editor:Yoshihiro Sawano

1, Department of Mathematics, Chaohu University, Hefei 238000, China

2, School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

3, Department of Mathematics, Anhui Normal University, Wuhu 241000, China

Received 22 September 2016; Accepted 21 November 2016; 9 January 2017

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

As we know, Hytönen [1] introduced nonhomogeneous metric measure spaces, which include both spaces of homogeneous type and nondoubling measure spaces as special cases. From then on, many results on nonhomogeneous metric measure spaces are obtained by many researchers. Hytönen et al. [2] and Bui and Duong [3] introduced independently the atomic Hardy space ^{H 1} ( μ ) and proved that the dual space of ^{H 1} ( μ ) is R B M O ( μ ) . The authors in [3] also proved that Calderón-Zygmund operator and commutators are bounded in ^{L p} ( μ ) for 1 < p < ∞ . Recently, some equivalent characterizations were established by Liu et al. [4] for the boundedness of Calderón-Zygmund operators on ^{L p} ( μ ) for 1 < p < ∞ . Fu et al. [5, 6] established boundedness of multilinear commutators of Calderón-Zygmund operators and generalized fractional integrals on Orlicz spaces, respectively. For more results, one can refer to [2, 4, 7-15] and the references therein.

θ -type Calderón-Zygmund operator was firstly introduced by Yabuta [16] in 1985. Later, many researchers further studied the properties of this operator. We [17] obtained the boundedness of θ -type Calderón-Zygmund operator and commutators on nondoubling measure spaces. Ri et al. [18, 19] researched the boundedness of θ -type Calderón-Zygmund operator on Hardy spaces with nondoubling measures and nonhomogeneous metric measure spaces, respectively. Zheng et al. [20, 21] studied the bounded properties for bilinear θ -type Calderón-Zygmund operator and maximal bilinear θ -type Calderón-Zygmund operator on nonhomogeneous metric measure spaces, respectively.

In this paper, we prepare to study the boundedness for commutators generated by bilinear θ -type Calderón-Zygmund operators and R B M O ( μ ) functions on nonhomogeneous metric measure spaces. And we obtain that these commutators are bounded on Lebesgue spaces, provided that bilinear θ -type Calderón-Zygmund operator is bounded from ^{L 1} ( μ ) × ^{L 1} ( μ ) to ^{L 1 / 2 , ∞} ( μ ) , where ^{L p} ( μ ) and ^{L p , ∞} ( μ ) denote the Lebesgue spaces and weak Lebesgue spaces, respectively. This result includes the corresponding results on both spaces of homogeneous type and nondoubling measure spaces. It is even new in the settings of spaces of homogeneous type and nondoubling measure spaces.

Throughout this paper, ^{L c ∞} ( μ ) denotes the set of all ^{L ∞} ( μ ) functions with compact support. C always denotes a positive constant independent of the main parameters involved, but it may vary from line to line. And ^{p [variant prime]} is the conjugate index of p ; namely, 1 / p + 1 / ^{p [variant prime]} = 1 . Now, let us recall some definitions and terminologies.

Definition 1 (see [1]).

A metric space ( X , d ) is geometrically doubling if there exists some _{N 0} ∈ N such that, for every ball B ( x , r ) ⊂ X , there exists a finite ball covering { B ( _{x i} , r / 2 ) _{} i} of B ( x , r ) such that the cardinality of this covering is at most _{N 0} .

Definition 2 (see [1]).

A metric measure space ( X , d , μ ) is upper doubling if μ is a Borel measure on X and there exists a function λ : X × ( 0 , + ∞ ) [arrow right] ( 0 , + ∞ ) and a constant _{C λ} > 0 such that, for every x ∈ X , r [...] ( x , r ) is nondecreasing, and for any x ∈ X , r > 0 , [figure omitted; refer to PDF]

Remark 3.

(i) Spaces of homogeneous type are upper doubling space, if we take λ ( x , r ) = μ ( B ( x , r ) ) . Also, nondoubling measure space, which satisfies the following polynomial growth condition: [figure omitted; refer to PDF] for all x ∈ ^{R d} and r > 0 , is also upper doubling measure space if we take λ ( x , r ) = C ^{r n} .

(ii) The authors [11] showed that there exists another function λ ~ such that, for any x , y ∈ X , d ( x , y ) <= r , [figure omitted; refer to PDF] Thus, one assumes that λ always satisfies (3) in this paper. As the singularity of commutators is stronger than that of bilinear operators, by [22], we suppose that there exists 0 < m < + ∞ , such that, for any a , r > 0 , x ∈ X , [figure omitted; refer to PDF]

Let α , β ∈ ( 1 , + ∞ ) ; a ball B ⊂ X is ( α , β ) -doubling if μ ( α B ) <= β μ ( B ) . As pointed in Lemma 2.3 of [3], there exist plenty of doubling balls with small radii and with large radii. In this paper, unless α and β are specified otherwise, one means ( α , β ) doubling ball is ( 6 , _{β 0} ) -doubling with a fixed number _{β 0} > m a x { ^{C λ 3 _{l o g 2} 6} , ^{6 n} } , where n = _{l o g 2 N 0} is the geometric dimension of the space.

Definition 4 (see [3]).

For two balls B ⊂ Q , let N B , Q be the smallest integer such that 6 N B , Q r B ≥ r Q ; we denote [figure omitted; refer to PDF]

Let θ be a nonnegative nondecreasing function on ( 0 , + ∞ ) satisfying [figure omitted; refer to PDF]

Definition 5.

A kernel K ( · , · , · ) ∈ ^{L l o c 1} ( ^{X 3} \ { ( x , _{y 1} , _{y 2} ) : x = _{y 1} = _{y 2} } ) is called the bilinear θ -type Calderón-Zygmund kernel if it satisfies the following:

(i) [figure omitted; refer to PDF]

for all ( x , _{y 1} , _{y 2} ) ∈ ^{X 3} with x ≠ _{y j} for j ∈ { 1,2 } .

(ii) [figure omitted; refer to PDF]

provided that C d ( x , ^{x [variant prime]} ) <= _{m a x 1 <= j <= 2} d ( x , _{y j} ) .

A bilinear operator _{T θ} is called bilinear θ -type Calderón-Zygmund operator with the above kernel K if for _{f 1} , _{f 2} ∈ ^{L c ∞} and x ∉ ^{[...] j = 1 2} s u p p _{f j} , [figure omitted; refer to PDF]

Remark 6.

As _{m a x 1 <= j <= 2} d ( x , _{y j} ) <= ^{∑ j = 1 2} d ( x , _{y j} ) <= 2 _{m a x 1 <= j <= 2} d ( x , _{y j} ) , (ii) in Definition 5 is equivalent to (ii[variant prime]) in the following statement:

(ii[variant prime]): [figure omitted; refer to PDF]

provided that C d ( x , ^{x [variant prime]} ) <= _{m a x 1 <= j <= 2} d ( x , _{y j} ) .

Remark 7.

(i) In [20, 21], the term [ ^{∑ j = 1 2} λ ( x , d ( x , _{y j} ) ) ^{] - 2} in (7) and (8) of this paper is substituted by _{m i n j ∈ { 1,2 }} [ λ ( x , d ( x , _{y j} ) ) ^{] - 2} . In fact, as [figure omitted; refer to PDF] and λ ( x , _{m a x j ∈ { 1,2 }} d ( x , _{y j} ) ) <= ^{∑ j = 1 2} λ ( x , d ( x , _{y j} ) ) <= 2 λ ( x , _{m a x j ∈ { 1,2 }} d ( x , _{y j} ) ) , Definition 5 in this paper is equivalent to Definition 1.4 in [20] or Definition 1.3 in [21]. Therefore, we can directly quote the result of Theorem 1.5 in [20] as Lemma 17 below in this paper.

(ii) Because we assume that _{T θ} is bounded from ^{L 1} ( μ ) × ^{L 1} ( μ ) to ^{L 1 / 2 , ∞} ( μ ) , it is enough to assume that K satisfies the regularity condition on the first variable, that is, (8) in this paper for getting the result of Theorem 10 below. For more details, one can refer to Remark 1.1 in [9].

Definition 8.

The commutator generated by bilinear θ -type Calderón-Zygmund operator _{T θ} and _{b 1} , _{b 2} ∈ R B M O ( μ ) is defined by [figure omitted; refer to PDF] Also, [ _{b 1} , _{T θ} ] and [ _{b 2} , _{T θ} ] are defined as follows, respectively: [figure omitted; refer to PDF]

Definition 9 (see [2]).

Let ρ > 1 be some fixed constant. A function b ∈ ^{L l o c 1} ( μ ) is said to belong to R B M O ( μ ) if there exists a constant C > 0 such that, for any ball B , [figure omitted; refer to PDF] and for any two doubling balls B ⊂ Q , [figure omitted; refer to PDF] where B ~ is the smallest ( α , β ) -doubling ball of the form ^{6 k} B with k ∈ N ∪ { 0 } , and _{m B ~} ( b ) is the mean value of b on B ~ : namely, [figure omitted; refer to PDF] The minimal constant C in (15) and (16) is the R B M O ( μ ) norm of b , which is denoted by _{( b ) [low *]} .

Theorem 10.

Let 1 < _{p 1} , _{p 2} < + ∞ , 1 / q = 1 / _{p 1} + 1 / _{p 2} , _{b 1} , _{b 2} ∈ R B M O ( μ ) . Assume that _{f 1} ∈ ^{L _{p 1}} ( μ ) , _{f 2} ∈ ^{L _{p 2}} ( μ ) with _{∫ X T θ} ( _{f 1} , _{f 2} ) ( x ) d μ ( x ) = 0 , _{∫ X} [ _{b 1} , _{T θ} ] ( _{f 1} , _{f 2} ) ( x ) d μ ( x ) = 0 , _{∫ X} [ _{b 2} , _{T θ} ] ( _{f 1} , _{f 2} ) ( x ) d μ ( x ) = 0 , _{∫ X} [ _{b 1} , _{b 2} , _{T θ} ] ( _{f 1} , _{f 2} ) ( x ) d μ ( x ) = 0 if ( μ ) < ∞ . If _{T θ} is bounded from ^{L 1} ( μ ) × ^{L 1} ( μ ) to ^{L 1 / 2 , ∞} ( μ ) , then there exists a constant C > 0 such that [figure omitted; refer to PDF]

Remark 11.

The result of Theorem 10 is still valid for commutators of multilinear θ -type Calderón-Zygmund operators with R B M O ( μ ) functions.

2. Preliminaries

For any f ∈ ^{L l o c 1} ( μ ) , the noncentered doubling maximal operator is defined by [figure omitted; refer to PDF] and the sharp maximal operator ^{M #} is denoted by [figure omitted; refer to PDF] where _{Δ x} [: =] ( ) ( B , Q ) : x ∈ B ⊂ Q and B , Q are doubling balls ( ) .

For any 0 < η < 1 , denote [figure omitted; refer to PDF]

Let ρ > 1 , p ∈ ( 1 , ∞ ) , and s ∈ ( 1 , p ) ; the noncentered maximal operator _{M s , ( ρ )} f is defined by [figure omitted; refer to PDF] When s = 1 , we simply write _{M 1 , ( ρ )} f ( x ) as _{M ( ρ )} f .

Lemma 12 (see [3, 11]).

(i) For any f ∈ ^{L l o c 1} ( μ ) and μ - a.e. x ∈ X , [figure omitted; refer to PDF]

(ii) If ρ ≥ 5 , then the operator _{M ( ρ )} f is bounded on ^{L p} ( μ ) for p > 1 and _{M s , ( ρ )} is bounded on ^{L p} ( μ ) for p > s > 1 .

Lemma 13 (see [3, 5]).

Assume that f ∈ ^{L l o c 1} ( μ ) with _{∫ X} f ( x ) d μ ( x ) = 0 if ( μ ) < ∞ . For 1 < p < ∞ and 0 < η < 1 , if inf [...] ( 1 , _{N η} f ) ∈ ^{L p} ( μ ) , then there exists a constant C > 0 such that [figure omitted; refer to PDF]

Lemma 14 (see [5, 23]).

Suppose that 1 <= p < ∞ and 1 < ρ < ∞ . Then b ∈ R B M O ( μ ) if and only if for any ball B ⊂ X , [figure omitted; refer to PDF] and for any two doubling balls B ⊂ Q , [figure omitted; refer to PDF]

Lemma 15 (see [5]).

For any k ∈ ^{N +} , [figure omitted; refer to PDF]

Lemma 16 (Kolmogorov's theorem).

Let ( X , μ ) be a probability measure space and let 0 < p < q < ∞ ; then there exists a constant C > 0 , such that _{( f ) ^{L p} ( μ )} <= C _{( f ) ^{L q , ∞} ( μ )} for any measurable function f .

Lemma 17 (see [20]).

Let 1 < _{p 1} , _{p 2} < + ∞ , 1 / q = 1 / _{p 1} + 1 / _{p 2} , _{f 1} ∈ ^{L _{p 1}} ( μ ) , and _{f 2} ∈ ^{L _{p 2}} ( μ ) . If _{T θ} is bounded from ^{L 1} ( μ ) × ^{L 1} ( μ ) to ^{L 1 / 2 , ∞} ( μ ) , then there exists a constant C > 0 such that [figure omitted; refer to PDF]

3. Proof of Main Result

Lemma 18.

Suppose that 0 < η < 1 / 2 , 1 < _{p 1} , _{p 2} , q < ∞ , 1 < s < q and _{b 1} , _{b 2} ∈ R B M O ( μ ) . If _{T θ} is bounded from ^{L 1} ( μ ) × ^{L 1} ( μ ) to ^{L 1 / 2 , ∞} ( μ ) , then there exists a constant C > 0 such that, for any x ∈ X , _{f 1} ∈ ^{L _{p 1}} ( μ ) , and _{f 2} ∈ ^{L _{p 2}} ( μ ) , [figure omitted; refer to PDF]

Proof.

Because ^{L c ∞} ( μ ) is dense in ^{L p} ( μ ) for 1 < p < ∞ , we only consider the situation of _{f 1} , _{f 2} ∈ ^{L c ∞} ( μ ) . Also, by Lemma 3.11 in [5], we can assume that _{b 1} , _{b 2} ∈ ^{L ∞} ( μ ) . As it has the similar method to estimate (29), (30), and (31), here we only estimate (29) for complicity.

To obtain (29), with the similar way to prove Theorem 9.1 in [24], it suffices to show that [figure omitted; refer to PDF] holds for any x ∈ B , and [figure omitted; refer to PDF] for any balls B ⊂ Q with x ∈ B , where B is an arbitrary ball and Q is a doubling ball. Denote [figure omitted; refer to PDF]

As [figure omitted; refer to PDF] thus [figure omitted; refer to PDF]

For I . Let _{s 1} , _{s 2} > 1 such that 1 / _{s 1} + 1 / _{s 2} + 1 / s = 1 / η . By Hölder's inequality and Lemma 14, [figure omitted; refer to PDF]

Let us estimate I I , let t > 1 such that 1 / t + 1 / s = 1 / η , and then [figure omitted; refer to PDF]

Similar to estimate I I , [figure omitted; refer to PDF]

Let us turn to estimate I V . Let ^{f j 1} = _{f j χ ( 6 / 5 ) B} , ^{f j 2} = _{f j} - ^{f j 1} for j ∈ { 1,2 } ; then [figure omitted; refer to PDF]

For I _{V 1} , let p = η and q = 1 / 2 such that 0 < η < 1 / 2 . Using Kolmogorov's theorem, Hölder's inequality, Lemma 14, and the boundedness from ^{L 1} ( μ ) × ^{L 1} ( μ ) to ^{L 1 / 2 , ∞} ( μ ) of _{T θ} , [figure omitted; refer to PDF]

To estimate I _{V 2} , by Definition 5, Lemmas 14 and 15, Hölder's inequality, and some properties of λ , [figure omitted; refer to PDF] Similarly, we obtain [figure omitted; refer to PDF]

For I _{V 4} , by (ii) of Definition 5 and some properties of λ , [figure omitted; refer to PDF]

Let us estimate I _{V 41} . With the help of the fact that [figure omitted; refer to PDF] by Lemmas 14 and 15 and Hölder's inequality, we have [figure omitted; refer to PDF] With the same method to estimate I _{V 41} , [figure omitted; refer to PDF]

Thus, taking the mean over _{z 0} ∈ B , we have [figure omitted; refer to PDF] So (32) can be obtained.

Next we prove (33). Denote N = _{N B , Q} + 1 ; then [figure omitted; refer to PDF]

Using the method to estimate I _{V 4} , [figure omitted; refer to PDF]

Let us estimate _{J 2} . As [figure omitted; refer to PDF] then [figure omitted; refer to PDF]

Let us estimate _{J 21} . Note that ( 6 / 5 ) Q ⊂ ^{6 N} B ; write [figure omitted; refer to PDF] Let us estimate _{E 1} firstly. Since Q is a doubling ball, we have [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] For _{E 2} , let [upsilon] > 1 and 1 < _{s 1} < _{p 1} such that 1 / [upsilon] = 1 / _{s 1} + 1 / _{p 2} ; denote 1 / _{s 1} = 1 / _{s 2} + 1 / _{p 1} . Note that Q is a doubling ball. By Lemmas 16 and 14 and Hölder's inequality, [figure omitted; refer to PDF]

For _{E 3} , by (i) of Definition 5, Lemma 14, and Hölder's inequality, [figure omitted; refer to PDF] Thus, [figure omitted; refer to PDF] Also, we have [figure omitted; refer to PDF]

By (26) in Lemma 14, [figure omitted; refer to PDF] _{J 22} and _{J 23} also have similar estimate of _{J 21} . Therefore, [figure omitted; refer to PDF]

Using the similar method to estimate I _{V 2} , [figure omitted; refer to PDF] Hence, (33) is proved. Thus, the result of Lemma 18 is proved.

Proof of Theorem 10.

Because the proof of the result of ( μ ) < ∞ is similar to that of ( μ ) = ∞ , now we only prove the result of ( μ ) = ∞ . Let 0 < η < 1 / 2 , 1 < _{p 1} , _{p 2} , q < ∞ , 1 / q = 1 / _{p 1} + 1 / _{p 2} , 1 < s < q , _{f 1} ∈ ^{L _{p 1}} ( μ ) , _{f 2} ∈ ^{L _{p 2}} ( μ ) , and _{b 1} , _{b 2} ∈ R B M O ( μ ) . By Lemmas 12-14 and 17 and Hölder's inequality, then [figure omitted; refer to PDF] Thus, the proof of Theorem 10 is finished.

Acknowledgments

This work was supported by Natural Science Foundation of Anhui Province (no. 1608085QA12) and Natural Science Foundation of Education Committee of Anhui Province (nos. KJ2015A117 and KJ2016A506).

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