Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 842375, 15 pages

http://dx.doi.org/10.1155/2013/842375

Research Article

Commutators of Higher Order Riesz Transform Associated with Schrödinger Operators

Yu Liu,¹ Lijuan Wang,¹ and Jianfeng Dong²

¹School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

²Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 25 February 2013; Accepted 3 April 2013

Academic Editor: Józef Banaś

Copyright © 2013 Yu Liu 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.

Abstract.

Let be a Schrödinger operator on (), where is a nonnegative potential belonging to certain reverse Hölder class for . In this paper, we prove the boundedness of commutators generated by the higher order Riesz transform , where , which is larger than the space . Moreover, we prove that is bounded from the Hardy space into weak .

1. Introduction

Let be a Schrödinger operator on , , where is a nonnegative potential belonging to the reverse Hölder class for some . In this paper, we will consider the higher order Riesz transforms associated with the Schrödinger operator defined by and the commutator

We also consider its dual higher order transforms associated with the Schrödinger operator defined by and the commutator

where , which is larger than the space .

Because the investigation of commutators of singular integral operators plays an important role in Harmonic analysis and PDE, many authors concentrate on this topic. It is well known that Coifman et al. [1] proved that is a bounded operator on for if and only if when is a Calderón-Zygmund operator. See [2, 3] for the research development of the commutator on Euclidean spaces and [4–6] on spaces of homogeneous type.

In recent years, singular integral operators related to Schrödinger operators and their commutators have been brought to many scholars attention. See, for example, [7–19] and their references. Especially, Guo et al. [12] investigated the boundedness of the commutators when . But their method is not valid to prove the boundedness of the commutators when . In fact, since

then may be written as follows:

where and . If , by using Corollary  1 in [12], we obtain the boundedness of . But if and , it follows from [1] that is not bounded on , and then we cannot obtain the boundedness of .

Motivated by [12, 15, 17], our aim in this paper is to investigate the estimates and endpoint estimates for when . Different from the classical higher order Riesz transform, there exist some new problems for the higher order Riesz transform . We need to obtain some new estimates for when the potential satisfies more stronger conditions.

A nonnegative locally -integrable function () is called to belong to if there exists a constant such that the reverse Hölder inequality

holds for every ball in .

Moreover, a locally bounded nonnegative function , if there exists a positive constant such that

holds for every in and .

Obviously, , if . But it is important that the class has a property of  ‘‘self-improvement’’; that is, if , then for some . Furthermore, it is easy to see that for any .

Assume that and . The Schrödinger operator generates a () semigroup . The maximal function with respect to the semigroup is given by

The Hardy space associated with the Schrödinger operator is defined as follows in terms of the maximal function mentioned earlier (cf. [20]).

Definition 1. A function is said to be in if the semigroup maximal function belongs to . The norm of such a function is defined by

We introduce the auxiliary function defined by

It is known that for any (from Lemma 8 in Section 2).

Definition 2. Let . A measurable function is called a -atom associated to the ball if and the following conditions hold:

(i) for some and ,(ii),(iii)when , .

The space admits the following atomic decompositions (cf. [21]).

Proposition 3. Let . Then, if and only if can be written as , where are -atoms and . Moreover,

where the infimum is taken over all atomic decompositions of into -atoms.

Following [17], the class of locally integrable function is defined as follows:

for all and , where and . A norm for , denoted by , is given by the infimum of the constants satisfying (11), after identifying functions that differ upon a constant. If we let in (11), then is exactly the John-Nirenberg space . Denote that . It is easy to see that for . Bongioanni et al. [17] gave some examples to clarify that the space is a subspace of .

Let be the auxiliary function of . Our main results are given as follows.

Theorem 4. Suppose that for some , (), , and . Let . The commutator is bounded on for , where .

By duality, we immediately have the following.

Corollary 5. Suppose that for some , (), , and . Let . The commutator is bounded on for .

Furthermore, we get the endpoint estimate for the commutator .

Theorem 6. Suppose that for some , (), , and . Let . Then, for any ,

Namely, the commutator is bounded from into .

This paper is organized as follows. In Section 2, we collect some known facts about the auxiliary function and some necessary estimates for the kernel of the higher order Riesz transform . In Section 3, we give the proof of Theorems 4 and 6. Section 4 gives the corresponding results when the potential satisfies stronger conditions. In Section 5, we give some examples for the potentials in Theorems 4 and 6.

Throughout this paper, unless otherwise indicated, we always assume that for some . We will use to denote the positive constants, which are not necessarily same at each occurrence even be different in the same line, and may depend on the dimension and the constant in (5) or (6). By and , we mean that there exist some constants such that and , respectively.

2. Some Lemmas

In this section, we collect some known results about auxiliary function and some necessary estimates for the kernel of the higher order Riesz transform in the paper.

Lemma 7. is a doubling measure; that is, there exists a constant such that

Especially, there exist constants and such that

holds for every ball and .

Lemma 8. There exist constants such that

In particular, if .

Using the Hölder inequality and condition, we have the following.

Lemma 9. Let

Moreover, if , then there exists such that

Lemma 10. For ,

There exist and such that

Let be the fundamental solution of . Then, there exists such that for each ,

In particular, is the fundamental solution of the Schrödinger operator . If , then there exists such that for each ,

The previous facts had been obtained by Shen in [8].

We denote the fundamental solution of by , which satisfies the following.

(i)There exists such that

(ii)There exists such that

Lemma 11. Suppose that for some and for some . Assume that in . Then,

Proof. Let such that on , , , , and . Since

then, for ,

Therefore, we have, for ,

where we use Lemma 9 and in Lemma 10 in the last step.

Therefore, we complete the proof of the lemma.

Furthermore, we get the following corollary via the proof of Lemma 11.

Corollary 12. Suppose that for some and for some . There exists a constant such that for each ,

Lemma 13. Suppose that for some and for some . There exists a constant such that for each ,

where for .

Proof. Let . Assume that . It follows from the embedding theorem of Morrey, Corollary  1, and Remark  4.9 in [8] that

where .

Similarly, we have the following.

Lemma 14. Suppose that for some and for some . There exists a constant such that for each ,

where for .

Corollary 15. Suppose that for some , for some , and there exists a constant such that

There exists a constant such that for each ,

Proof. Since for , then by using Lemma 8,

Therefore, by Corollary 12,

Furthermore, we obtain the following corollary by using Corollary 12 and Lemma 14.

Corollary 16. Suppose that for some , for some and satisfies (32). There exists a constant such that for each ,

where for .

Remark 17. Following Remark 5 in [22], we know that if is a nonnegative polynomial, condition (32) holds true. Therefore, Corollaries 15 and 16 also hold true.

Lemma 18. Suppose that for some , and .

(1) and are bounded on the space , where .(2) is bounded on the space for .(3) is bounded on the space for .

Since , by using in Lemma 18, we obtain the following.

Lemma 19. Suppose that for some . Then, for any ,

2.1. Some Lemmas Related to BMO Spaces

In this section, we recall some propositions and lemmas for the BMO spaces in [17].

A ball is called critical. In [20], Dziubański and Zienkiewicz gave the following covering lemma on .

Proposition 20. There exists a sequence of points in , such that the family of critical balls , , satisfies the following:

(i);(ii)there exists such that for every , .

Lemma 21. Let and . If , then

for all , with and , where and is the constant appearing in Lemma 8.

Lemma 22. Let , , and . Then,

for all , with as in (38).

Given that , we define the following maximal functions for and :

where .

Also, given a ball , for and , we define

where .

Lemma 23. For , there exist and such that if is a sequence of balls as in Proposition 20, then

for all .

3. Proofs of the Main Results

Firstly, in order to prove Theorem 4, we need the following lemmas. As usual, for , we denote by the -maximal function which is defined as

Lemma 24. Suppose that for some , (), , and . Let . Then, there exists a constant such that

for all for and every ball .

Proof. Let and . Write as

Firstly, by the Hölder inequality with and Lemma 21,

If we write with , then

where we use the fact that is bounded on with .

By Corollary 12 and the Hölder inequality, we have

where

For , note that by using Lemma 8. We also note that . Then,

Since , then

Using the Hölder inequality and the boundedness of the fractional integral with , we have

Since , we obtain

where we use the assumption that and in Lemma 10.

We also have

Therefore, using the fact that , we obtain

where we choose large enough such that the previous series converges and we use the fact that .

To deal with the second term of (45), we split again with .

Firstly, using the Hölder inequality and the boundedness of on ,

where , , and we have used Lemma 21 in the last inequality.

For the remaining term, we firstly see the fact that and . Then, we deal with

where

By the Hölder inequality and Lemma 22, we have

where , and we choose large enough. The following estimate is similar to the estimate of . We repeat the previous method.

Then,

Using the Hölder inequality and the boundedness of the fractional integral with , we have

Since , we have already obtained

Also,

where .

Therefore, using that , we obtain

where we choose large enough such that the previous series converges and we use the fact that .

Therefore, this completes the proof.

Remark 25. Similarly, we can conclude that the previous lemma also holds if the critical ball is replaced by .

Lemma 26. Suppose that for some , (), , and . Let . Then, there exists a constant such that

for all for and , with , where .

Proof. Denote that . Note that and . By the estimate (29), we have

For , by using the Hölder inequality and Lemma 22, we have

where is the least integer such that .

To deal with , using Lemma 22 and choosing , we have

where we use the fact that when .

To deal with , by using Lemma 22 and ,

where and .

Since , then

for all , where we use the fact that .

Therefore,

where we use that .

At last, for we have, for ,

where and .

Furthermore, by using Lemma 7,

where we use the fact that . Consider

where we choose large enough such that the previous series converges and we use the fact that .

Proof of Theorem 4. We start with a function for . By Lemmas 24 and 26 and Remark 25, we have

where we use the finite overlapping property given by Proposition 20 and the boundedness of in for .

Next, we consider the term . Our goal is to find a pointwise estimate of . Let and , with such that . If , with , then we write

Therefore, we need to control the mean oscillation on of each term that we call , , and . By using the Hölder inequality and Lemma 21, we obtain

since .

To estimate , let . Then,

where .

For , by Lemma 26, we obtain

Therefore, we have proved that

Then, we have obtained the desired result.

Proof of Theorem 6. For , we can write , where each is a atom and . Suppose that with . Write

Using the Hölder inequality, the boundedness of with , and Lemma 21,

since .

When we consider the term , we note that . Consider

Note that and

Then, by Lemma 22,

where we choose large enough.

Similarly,

where we choose large enough and we use the fact that .

Therefore, if , then

For , by using the vanishing condition of and Lemma 14, then

First of all, we need to obtain the following new estimate:

where we use the assumption that and in Lemma 10. Consider

where we use the fact that , , and we choose .

Secondly,

where we use the fact that .

Therefore, if , then

Thus, we have

Note that

where .

By the weak boundedness of (cf. Lemma 19), we get

Therefore,

This completes the proof of Theorem 6.

4. Another Case

In this section, we obtain same results for the commutator if we impose another condition on . Via Corollaries 15 and 16 in Section 2 and Theorems  1.2 and  1.3 in [23], we obtain the following theorems.

Theorem 27. Suppose that for some , for some , and satisfies (32). Let . The commutator is bounded on for .

Theorem 28. Suppose that for some , for some , and satisfies (32). Let . Then, for any ,

Namely, the commutator is bounded from into .

Remark 29. Following Remark 5 in [22], we know that if is a non-negative polynomial, condition (32) holds true. Furthermore, we know that if , where is a polynomial and , condition (32) holds true (see Remark 6 in [24]).

5. Examples

In this section, we give some examples for the potentials which can satisfy the assumption in Theorems 4 and 6. We always assume that throughout this section. Denote the norm of by .

Example  1.  Let

Following [25], we know that if is a polynomial of degree and , then belongs to . For , it is easy to see that . Moreover, it follows from (0.14) in [26] that

Therefore, .

If , then

Thus,

Therefore, . Clearly, . So, . Therefore, . Also, since , then . Then, the potential satisfies the assumption of Theorems 4 and 6.

Example  2.  Let . By the previous argument, we conclude that .

Then,

Thus, . Clearly, . From (99), we know that and . Therefore, . Also, since , then . Then, the potential satisfies the assumption of Theorems 4 and 6.

Example  3.  Let . By the previous argument, we conclude that .

Then,

Thus,

From (99), we know that . Since

then

Thus, . From (99), we know that . Therefore, . Also, since , then . Then, the potential satisfies the assumption of Theorems 4 and 6.

Acknowledgments

This paper is supported by Research Fund for the Doctoral Program of Higher Education of China under Grant (no. 20113108120001), the Shanghai Leading Academic Discipline Project (J50101), the National Natural Science Foundation of China under Grant (no. 10901018), and the Fundamental Research Funds for the Central Universities.

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