Compactness of the Commutator of Multilinear Fourier Multiplier Operator on Weighted Lebesgue Space

Abstract

Let be the multilinear Fourier multiplier operator associated with multiplier satisfying the Sobolev regularity that for some . The authors prove that if and , then the commutator is bounded from to . Moreover, the authors also prove that if and , then the commutator is compact operator from to .

1. Introduction

The study of the multilinear Fourier multiplier operator was originated by Coifman and Meyer in their celebrated work [1, 2]. Let ; the multilinear Fourier multiplier operator is defined by for all , where and is the Fourier transform of . Coifman and Meyer [2] proved that if satisfies for all with , then is bounded from to for all , with . For the case of , Kenig and Stein [3] and Grafakos and Torres [4] improved Coifman and Meyer’s multiplier theorem to the indices by the multilinear Calderón-Zygmund operator theory. In the last several years, considerable attention has been paid to the behavior on function spaces for when the multiplier satisfies certain Sobolev regularity condition. Let satisfy For , set Tomita [5] proved that if for some , then is bounded from to provided that and . Grafakos and Si [6] considered the mapping properties from to for when . Let satisfy the Sobolev regularity that where . Miyachi and Tomita [7] proved that if for some , then is bounded from to provided that with .

As well known, when satisfies (3) for some , then is a standard multilinear Calderón-Zygmund operator, and then by the weighted estimates with multiple weights for multilinear Calderón-Zygmund operators, which were estimated by Lerner et al. [8], we know that, for any and with and weights such that , By a suitable kernel estimate and the theory of multilinear singular integral operator, Bui and Duong [9] established the weighted estimates with multiple weights for when satisfies (3) for and . Hu and Yi [10] considered the behavior on for when satisfies (5) for and showed that enjoys the same mapping properties as that of the operator .

Now, considerable attention has been paid to the behavior on the compactness of multilinear Fourier multipliers operator with Sobolev regularity. Let be the closure of in the topology, which coincides with the space of functions of vanishing mean oscillation (see [11, 12]). Bényi and Torres [13] proved that if and is multilinear Calderón-Zygmund operator, then, for with , the commutator is compact operator from to . Hu [14] proved that if is a multilinear multiplier which satisfies (5) for some , , , and for and with , then is compact operators from to . Bényi et al. [15] proved that if and is multilinear Calderón-Zygmund operator, , then, for with , the commutator is compact operator from to .

Inspired by the above, we consider the weighted compactness of the commutator of the multilinear Fourier multiplier operator on .

Given a multilinear Fourier multiplier operator , and , the commutator is defined by with

Our main results are stated as follows.

Theorem 1. Suppose that be a multilinear multiplier which satisfies (7) for some and . Let , for and with . If the weights satisfy , then, for any , is bounded from to .

Theorem 2. Suppose that be a multilinear multiplier which satisfies (7) for some and . Let , for and with . If the weights satisfy , then, for any , is a compact operator from to .

Because the regularity condition is stronger than , we have the following corollaries.

Corollary 3. Suppose that be a multilinear multiplier which satisfies (5) for some . Let , for and with . If the weights satisfy , then, for any , is bounded from to .

Corollary 4. Suppose that be a multilinear multiplier which satisfies (5) for some . Let , for and with . If the weights satisfy , then, for any , is a compact operator from to .

The paper is organized as follows. In Section 2, we give some necessary notion and lemmas. In Section 3, we prove our main results, Theorems 1 and 2. Throughout the paper, always denotes a positive constant that may vary from line to line but remains independent of the main variables. We use the symbol to indicate that there exists a positive constant such that . We use to denote a ball centered at with radius . For a ball and , we use to denote the ball concentric with whose radius is times of . As usual, denotes the Lebesgue measure of a measurable set in and denotes the characteristic function of . For , we denote by the dual exponent of .

2. Some Notations and Lemmas

Let us first introduce some definitions below.

Definition 5. Let be an integer, and let be weights, , , with , . Set and . Then

Definition 6. For any , and , is defined by For , is the maximal function The sharp maximal function of Fefferman-Stein is defined by where .

Next, we give some symbols.

Let and satisfy (3). For , define Then where denotes the inverse Fourier transform of .

For , let and denote by the multiplier operator associated with . It is obvious that is an -linear singular operator with kernel For an integer with and , let

Assume that is a multilinear operator initially defined on the -fold product of Schwartz spaces, and, taking values in the space of tempered distributions, By the associated kernel , we mean that is a function defined off the diagonal in , satisfying for all functions and all . It is easy to see that the associated kernel to Fourier multiplier operator is given by

To prove main results, we need the following lemmas.

By the reverse Hölder inequality, we have the first lemma.

Lemma 7. Assume that , with , , and with . Let ; then there exists a constant such that .

For and , the weighted Lebesgue space of mixed type is defined by the norm where .

Lemma 8 (see [16]). Let , , and for . Then there exists a constant such that for all with .

The following lemma is the key to our main lemma.

Lemma 9. Suppose that be a multilinear multiplier which satisfies (7) for some . Let , , , and is the same as that appears in Lemma 7. Then for all with for , where .

Proof. By Lemma 8, ; then . Fix a point and a cube such that . It suffices to prove for some constant . We decompose with for all and . Then where . Then we can write Applying Kolmogorov’s inequality to , we have since is bounded from to .
Take We claim that, for any , Let At first, we consider the case . We get Denote , , and the side length of a cube ; it follows from Lemma 7 that Given that , we have that On the other hand, a similar process follows that in [17]; we get that where . Since we have From Lemma 8, . It is deduced that So It remains to consider the case that there exists a proper subset of , , such that . We have By the same argument as that of the case , we have that This completes the proof.

Lemma 10. Suppose that be a multilinear multiplier which satisfies (7) for some . Let , , and , and is the same as that appears in Lemma 7. Then, for and any , that is, , there exists some constant such that for all -tuples of bounded measurable functions with compact support.

The proof of the above lemma is standard. A statement similar to Lemma 2.7 in [17] with minor modifications deduces the estimates. We omit the details here.

Lemma 11. Let be a multilinear multiplier which satisfies (7) for some , such that . Then, for every , and ,

By a similar way in the proof of the Lemma 2.4 in [14], with slight changes, we can get the conclusion of Lemma 11 and we omit the details.

About the proof of compactness, as in [18] we will rely on the Fréchet-Kolmogorov theorem characterizing the precompactness of a set in . More precisely, see Yosida’s book [19]. For more about compactness, we refer to [20, 21].

Lemma 12. A set is precompact in , if and only if(i),(ii) uniformly in ,(iii) uniformly in .

3. Proof of Theorems 1 and 2

Proof of Theorem 1. We only present the case that . We have by Lemma 10 and Theorem 3.2 in [8] which completes the proof of Theorem 1.

Proof of Theorem 2. We will employ some ideas of Bényi and Torres [13]. Without loss of the generality, we only prove the case . Let , , with , and . Note that, for any and almost every ,
It is enough to prove that the following conditions hold:(a) is bounded from to ;(b)for each fixed , there exists a constant which is independent of , , and such that (c)for each fixed , there exists a constant which is independent of , , and such that, for all with , Then by the Fatou Lemma, the conditions (a), (b), and (c) still hold true if is replaced by .
It is clear that the first condition (a) holds according to Theorem 1.
Then, we prove the conclusion (b). Let be large enough such that and let . Then for every with , we have by Lemma 11 that if we choose and in Lemma 11 respectively. Therefore, where the last inequality holds by the fact (see [22, 23]) that for , This in turn leads to conclusion (b) directly.
We turn our attention to conclusion (c). We write with with a convenient choice to be determined later.
In a completely same way in the proof of Theorem  1.1 in [14], we can obtain the estimate of . We only list the results and omit the details.
Fix each , set with and for each , where constant . Our estimates for those terms of then lead to that when , this establishes conclusion (c) and we conclude that is compact.
In a completely analogous way, if , then is compact. Moreover, then is compact, thus we complete the proof of Theorem 2.

Conflict of Interests

The authors declare that they do not have any commercial or associative interest that represents a conflict of interests in connection with the work submitted.

Acknowledgments

The authors would like to thank the referee for his/her helpful suggestions. The paper is supported by the National Natural Science Foundation of China (11261055), the Natural Science Foundation Project of Xinjiang (2011211A005), and Xinjiang University Foundation Project (BS120104).