On the Compactness of the Commutator of the Parabolic Marcinkiewicz Integral with Variable Kernel

Chen, Yanping; Zhu, Kai; Ding, Yong

doi:https://doi.org/10.1155/2014/693534

Journal of Function Spaces

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Research Article | Open Access

Volume 2014 | Article ID 693534 | https://doi.org/10.1155/2014/693534

On the Compactness of the Commutator of the Parabolic Marcinkiewicz Integral with Variable Kernel

Yanping Chen,¹Kai Zhu,¹and Yong Ding²

Academic Editor: Gestur Ólafsson

Received30 Jun 2014

Accepted17 Oct 2014

Published12 Nov 2014

Abstract

The authors prove that the commutator of the parabolic Marcinkiewicz integral with variable kernel is a compact operator on if and only if . The result is substantial improvement and extension of some known results.

1. Introduction and Main Results

Let be the -dimension Euclidean space. Let be the unit sphere on equipped with the Lebesgue measure . For , denote by the multiplication operator defined by for the measurable function . Suppose that is a linear operator on some measurable function space; then the commutator formed by and is defined by A famous theorem of Coifman et al. [1] characterized the -boundedness of , where () are the Riesz transforms and . Using this characterization, the authors of [1] got a decomposition theorem of the real Hardy space . Uchiyama [2] and Janson [3] showed that the Riesz transform may be replaced by the Calderón-Zygmund singular integral operator . Moreover, in 1978, Uchiyama [2] gave a characterization of the compactness for the commutator defined by where is homogeneous function of degree zero on , that is, and has mean zero on , the unit sphere in , that is,

Theorem A (see [2]). Suppose that satisfies (3) and (4). Let .(i)If is a compact operator on for some , , then .(ii)If , then is a compact operator on for , where is the BMO-closure of and denotes the set of -functions with compact support in .

The interest in the compactness of in complex analysis is from the connection between the commutators and the Hankel-type operators. In fact, Beatrous and Li [4] proved the boundedness and compactness of on over some spaces of homogeneous type. Krantz and Li (see [5]) have applied the characterization of -compactness of commutator to give a compactness of Hankel operators on holomorphic Hardy spaces , where is a bounded, strictly pseudoconvex domain in .

In this paper, we will study the compactness of the commutator of the parabolic Marcinkiewicz integral with variable kernel. Let us recall the definition of parabolic metric. Let be fixed real numbers, . For fixed , the function is a decreasing function in (). We denote by the unique solution of the equation . In 1966, Fabes and Rivière [6] proved that is a metric on . For and , define the dilation on by Then it is immediate to see that and when . One has the polar decomposition with and , where is a function on and bounded below uniformly by 1.

In 1974, Madych [7] gave the () boundedness of the parabolic Marcinkiewicz integral with respect to the transform , where with satisfying , and for . Inspired by the works in [6, 7], recently, Xue et al. [8] improved the above result. More precisely, the authors in [8] proved that the parabolic Marcinkiewicz integral is still bounded on if replacing by a kernel function with () satisfying the following conditions: Afterwards, many authors studied the properties of the parabolic Marcinkiewicz integral and its commutators which is defined by (see, e.g., [9–12]).

Notice that if , then and . In this case, and are just the classical Marcinkiewicz integral and it is commutators which play a very important role in the singular integral operators theory, function spaces theory, and PDE (see, e.g., [13–23]).

Let us turn to the definitions of the parabolic Marcinkiewicz integral with variable kernel and its commutator. Suppose that satisfies the following conditions: Then the parabolic Marcinkiewicz integral with variable kernel is defined by Let us now give the definition of parabolic BMO space. Denote by the ellipsoid center at and radius . More precisely, . For , we denote to be the -times extension of with the same center. That is, . Moreover, stands the Lebesgue measure of , which is comparable to . denotes the complement of .

For a measurable and locally integrable function set where is any ellipsoid in of radius and is the average of on . We say if (see [24]).

Let ; the commutator of the parabolic Marcinkiewicz integral with variable kernel is defined by

Notice that if , then ; and are the classical Marcinkiewicz integral with variable kernel and the classical commutator of Marcinkiewicz integral with variable kernel which was studied by many authors (see, e.g., [25–28]).

In 2009, Chen and Ding [29] studied the parabolic Marcinkiewicz integral with variable kernel and its commutator on the Lebesgue space , which is stated as follows.

Theorem B. Let . Suppose that satisfies (9), (10) and for every multi-index , Let . Then there exists a constant such that, for any ,

In this paper, we will give a characterization of the commutator of the parabolic Marcinkiewicz integral with variable kernel which is compact operator on (). First, we will prove that the commutator of the parabolic Marcinkiewicz integral with variable kernel is bounded on (). To show our result, we give the following definition.

Definition 1 (see [30]). Let and be Banach spaces and let be a subset of . Then is compact if is continuous and maps bounded subsets of into strongly precompact subsets of .

Our first result can be stated as follows.

Theorem 2. Let satisfy (9), (10), (14) and for any multi-index If , where is the -closure of and denotes the set of -functions with compact support in , then is a compact operator on ().

On the other hand, we will give the converse part of the compactness of the commutator of the Marcinkiewicz integral with variable kernel.

Theorem 3. Suppose that satisfies (9) and (10), and there exists such that, for any , where is the measure on which is induced from the Lebesgue measure on . If is a bounded operator on for some , then .

Theorem 4. Let satisfy the same conditions as in Theorem 2. If is a compact operator on for some (), then .

Remark 5. Our results are still new even for the case .

This paper is organized as follows. First, in Section 2, we give some important notations and lemmas, which will be used in the proofs of the main results. In Section 3, we prove Theorem 2. The proof of Theorem 3 is given in Section 4. In Section 5, we will give the proof of Theorem 4.

The letter in the paper denotes positive constants independent of essential variables. For , denotes the dual exponent of , that is, .

2. Notations and Lemmas

For a given measurable function , the Hardy-Littlewood maximal operator and the sharp maximal operator related to ellipsoids are defined by respectively. Define also the operator for .

Denote by the space of all -dimensional spherical harmonic of degree with its dimension . By [24, 31], we have

Furthermore, let be an orthonormal base of . Then () is a complete orthonormal system in and for any multi-index (see [24, 31]), If, for instance, , then is the Fourier series expansion of with respect to , where and (see [24, 31]) where , for any integer . In particular, the expansion of into spherical harmonics converges uniformly to . The next results will be employed in the forthcoming considerations.

Lemma 6 (see [29]). Suppose that . Let . Then for each and , where .

Lemma 7 (Frechet-Kolmogorov, see [32, 33]). A subset of , , is strongly precompact if and only if it satisfies the conditions

Lemma 8 (see [29]). Suppose that . Let . Then for each and . (Here and below, for and , denotes the -times extension of with the same center, that is,

Lemma 9 (see [32]). If , then where .

Lemma 10 (see [2, 32]). If , , is an ellipsoid centered at and radius , then there exist positive constants , , and (depending on , , and ), such that

Lemma 11 (see [34]). Suppose that is a measurable function, , and is a measurable set. Define for ; then

Lemma 12 (see [2, 32]). Let . Then if and only if satisfies the following three conditions:(i),(ii),(iii), for each , where .

It is easy to get the following lemma.

Lemma 13. If satisfies conditions (9) and (17), then for , for .

3. Proof of Theorem 2

Suppose that . Then, by Theorem A, we can get Then we can easily get is continuous on , so by Definition 1, it suffices to prove that, for any bounded set in , is strongly precompact in . Since for any there exists such that , then by (33), Thus, it suffices to prove that is strongly precompact in for . By Lemma 7, we need only to verify that (25)–(27) hold uniformly in .

Suppose that for every . Notice that and applying (33), we have On the other hand, for any , we may take large enough such that Assume for some . Thus, for any satisfying and every , then by and ~, we have Applying (36), we have Equation (38) shows that (27) holds uniformly in . Finally, to finish the proof of Theorem 2, it remains to show (26) holds uniformly in . We need to prove that, for any , if is sufficiently small, then for every , To do this, for , by (10), we have where So we get In this way, (14) and (23) imply for any for any integer . We fix hereafter . Let

It is easy to see that has the following properties: Moreover, . Then by the result in [11, 12], we know for , where is independent of .

For any , by the Minkowski inequality, the Hölder inequality, we get where Then by (43) and the Minkowski inequality, we get We first give the norm of . By (16), we have Moreover, by (47), we get Therefore, using and the Minkowski inequality, we get Now, we give the norm of . Take an arbitrary and such that . Decompose as We will estimate each term, respectively. For , when , by Lemma 8, (21), , , and the Minkowski inequality, we have Then by the Minkowski inequality, we get Similar to the estimate of , we can get Then by the Minkowski inequality, we get For , since , by Lemma 6, the Minkowski inequality, and Lemma 8, we have Then similar to estimate (56), we get About , we have Similar to the estimate of in [35], we get for any fixed , where is independent of and . So, for any , taking in (62), then by (43) and the boundedness of , we get Since , we have . Then by (63), we get For , since , and when , then by (21) and the Minkowski inequality, we get Then we can get Similarly, we can get Combining the estimate of with , by the Minkowski inequality and , we get which combined the norm of (see (53)); then by taking sufficiently small depending on , we can get Thus we establish the proof of Theorem 2.

4. Proof of Theorem 3

In this proof for is a positive constant depending only on , , , , and (). Suppose that is a bounded operator from to ; we are going to prove that . Without loss of generality, we can assume . To prove that , we need to show there exists a constant such that for any and , where . Since , we may assume . So it suffices to prove that for any fixed , where . Let where . We get . Then satisfies the following properties: Then for and , For , by (18), we know Noting that if and , by Lemma 9, we get . Since if , we can get . Then by (74), (76), (77), and the Hölder inequality, we get By (73), (74), (75), and , then by Lemma 13 and Lemma 8, we have Then using the same argument of the proof of Theorem 1.2 in [29], we get .

5. Proof of Theorem 4

Since is a compact operator on , then is bounded on . Thus from Theorem 3, . Thus we may assume . By Lemma 12, to prove that , it suffices to show that satisfies conditions (i), (ii), and (iii) of Lemma 12.

First, suppose that does not satisfy (i) of Lemma 12. Then there exists and a sequence of ellipsoids with , such that for every Let , where . So . Then has the following properties: for . It is easy to see that , where is independent of .

Let . If , then Similar to the proof of in Theorem 3, by (84), (85) and noting that , we get On the other hand, for , by (84), (87), , and when , we have On the other hand, for , by (84), (86), (87), and , we have In fact, in the above estimate we have used Lemma 13 for . Then similar to the proof of Theorem 1.3 in [29], we show that satisfies (i)–(iii) of Lemma 12.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Y. Chen is supported by NSF of China (Grant no. 11471033), NCET of China (Grant no. NCET-11-0574), and the Fundamental Research Funds for the Central Universities (FRF-TP-12-006B). Y. Ding is supported by NSF of China (Grant no. 11371057), SRFDP of China (Grant no. 20130003110003).

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Copyright © 2014 Yanping Chen et al. 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.

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