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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 281562, 10 pages
http://dx.doi.org/10.1155/2013/281562
Research Article

Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

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

Received 14 May 2013; Accepted 1 October 2013

Academic Editor: Baoxiang Wang

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, where is the laplacian on and the nonnegative potential belongs to the reverse Hölder class for some . Assume that . Denote by the weighted Hardy space related to the Schrödinger operator . Let be the commutator generated by a function and the Riesz transform . Firstly, we show that the operator is bounded from into . Secondly, we obtain the endpoint estimates for the commutator . Namely, it is bounded from the weighted Hardy space into .

1. Introduction

Let be a Schrödinger operator, where is the laplacian on and the nonnegative potential belongs to the reverse Holder class for some and . In this paper, we consider the Riesz transform associated with the Schrödinger operator as follows:

Let be a locally integrable function on and let be a linear operator. For a suitable function , the commutator is defined by . It is well known that when is a Calderón-Zygmund operator, Coifman et al. [1] proved that is a bounded operator on for if and only if .

Recently, some scholars have investigated the boundedness of the commutators generated by a function and Riesz transforms associated with the Schrödinger operator (cf. [28]). It follows from [9] that Riesz transform associated with the Schrödinger operator is not a Calderón-Zygmund operator if the potential . Their results imply that the boundedness of the commutators of Riesz transform associated with the Schrödinger operator depends on the nonnegative potential . In [5], the authors have obtained the weighted , , and weak the estimates for the commutator . In this paper we are interested in the weighted Hardy space estimates for , which are also the weighted endpoint estimates. It is noted that our main results generalize Theorem 2.7 and Theorem  4.1 in [3] to the weighted case and the function that we consider belongs to a larger class than the classical space.

Note that a nonnegative locally integrable function on is said to belong to if there exists such that the reverse Hölder inequality holds for every ball in . Obviously, if . But it is important that the class has a property of “self- improvement”; that is, if , then for some .

Assume for some . Then the auxiliary function introduced by Shen in [9] is defined as follows:

In [4], the authors define the class of locally integrable function such that

for all and , where and . A norm for , denoted by , is given by the infimum of the constants satisfying (4), after identifying functions that differ upon a constant. If we let in (4), then is exactly the John-Nirenberg space . Denote . It is easy to see that for . Hence, .

Throughout this paper, we set for any subset . Assume that the nonnegative function . We say that for if there is a constant such that for all balls in , where .

We say that , if there exists a positive constant , such that where is a Hardy-Littlewood operator.

Given a weight function for , as usual we denote by the space of all measurable functions satisfying When , will be taken to mean and . Moreover, denote by the space of all measurable functions satisfying

In the rest of this paper, we always assume that .

Because and , the Schrödinger operator generates a contraction semigroup . The maximal function associated with is defined by .

The Hardy space associated with the Schrödinger operator is defined in terms of the maximal function mentioned above (cf. [10]). Recently, the weighted Hardy space has been established by Liu et al. [11] and it is defined as follows.

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

Definition 2. Let . A function is called an -atom if and the following conditions hold:(i),(ii),(iii)if , then .
In [11], Liu et al. gave the following atomic decomposition for the space .

Proposition 3. Assume for some . Let . Then if and only if can be written as , where are -atoms, , and the sum converges in the quasinorm. Moreover, where the infimum is taken over all atomic decompositions of into -atoms.

Following the above definition of atoms and the above atomic decomposition, we know that the weighted Hardy space is not the special case of Hardy spaces established by Yang and Zhou in [12].

Now we are in a position to give the main results in this paper.

Theorem 4. Let . Suppose for . Then,

Theorem 5. Let . Then, for any ,
where . Namely, the commutator is bounded from into .

This paper is organized as follows. In Section 2, we recall some basic facts to prove main results in this paper. Section 3 gives the proof of weighted estimates of Riesz transform associated with the Schrödinger operator. In Section 4, we prove Theorem 5.

Throughout this paper, the letter stands for a constant and is not necessarily the same at each occurrence. By , we mean that there exists a constant such that . Moreover, for the ball , we denote the ball by , where is a positive constant.

2. Preliminaries

Firstly, we recall some lemmas of the auxiliary function which have been proved by Shen in [9]. Throughout this section we always assume for some .

Lemma 6. The measure satisfies the doubling condition; that is, there exists such that
holds for all balls in .

Lemma 7. For ,

Lemma 8. If , then
Moreover,

Lemma 9. There exists such that, for any and in ,
In particular, if .

Lemma 10. There exists such that

For the proofs of Lemma 6 to Lemma 10, readers can refer to [9].

A ball is called critical. Due to Lemma  2.3 in [10], we have Lemma 12 on .

Proposition 11. 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 , card .

Secondly, we recall the estimates of the fundamental solution of the operator and give the estimates of kernels of Riesz transform . Let denote the fundamental solution for the operator , where . Clearly, .

Lemma 12. Let be an integer.(1)Suppose for . Then there exists such that, for , (2)Suppose for . Then there exists such that, for ,

By the functional calculus, we may write

Let . From , it follows that where

And the adjoint operator of is given by where

Lemma 13. Suppose for some . Then there exists , for any integer , and ,

Lemmas 12 and 13 have been proved in [9] and in [2], respectively.

Thirdly, we recall some important and useful properties of weights (cf. Chapter V in [13]).

Proposition 14. (i) Let , . Then there exist a constant and depending only on and the constant of , such that for any ball .
(ii) Let . If , then .

At last, we review some basic facts about the space , which have been proved in [4].

Proposition 15. Let and let . If , then
for all , with and , where and is the constant appearing in Lemma 9.

Lemma 16. Let , let , and let ; then
for all , with as in (27).

3. Weighted Estimates of Riesz Transforms Associated with the Schrödinger Operators

In this section, we need to prove the weighted estimates of the Riesz transform associated with the Schrödinger operator, which will be used in the proof of Theorem 5.

Theorem 17. Let . Suppose for some . Then, for ,
where and .

Proof. By the improvement of “”, for . It follows from the proof of Theorem  0.5 in [9] that where and is the kernel for the operator .
Since , then it follows from Proposition  7.2 in [14] that for any .
By [13, Theorem 1, page 201] and [13, Corollary 2, page 205]
Since , then

By duality, we have the following.

Corollary 18. Let . Suppose for some . Then, for ,where .

Proof. It follows from (ii) in Proposition 14 that for , where . By using Theorem 17, we know that Therefore, by duality and (34), This completes the proof of the above corollary.

In order to prove Theorem 4, we need the following lemmas.

Lemma 19 (see [11, Lemma 3.12]). Let . Then there exists a constant , such that
By using the proof of Lemma  3.15 in [11], we immediately have the following.

Lemma 20. Let . Then there exists a constant , such thatwhere is the kernel of the operator .

Now we give the proof of Theorem 4.

Proof of Theorem 4. We show Theorem 4 by a method similar to the one used in the proof of Theorem 2 in [15]. By the Calderón-Zygmund decomposition in the proof of Theorem 3.5 on page  413 in [16], given and , we have , with , such that the following hold.(1), for a.e. .(2)Each is supported in a ball , (3).
Because , by Corollary 18, we know that is bounded on ; it is clear that
Let and . Then
We only need to consider for . If , then for any . By Lemma 19, we get
If , then for any . Since is a Calderón-Zygmund kernel, by Lemma 19, Lemma 20, and [16, Lemma 3.3, page 413], we obtain
Then Therefore
Theorem 4 is proved by combination of (39), (40), and (44).

4. Proof of Theorem 5

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

Using Hölder inequality, the weighted boundedness of , Propositions 14 and 15, for and with , we have

since .

When we consider the term , we note that as following:

Note that and

Then by Lemma 16 and Proposition 14, for any , where we have chosen large enough.

Similarly, for and , using Hölder inequality, Proposition 14, and the boundedness of the fractional integral , we obtain, for any , where we have chosen large enough and we have used Lemma 10 in the sixth inequality above.

Therefore, where we have used the fact that .

Thus, if , then

For , by using the vanishing condition of and Lemma 13, then First of all, we need to obtain the following estimate: where we have used Lemma 10 in the last inequality above.

Similarly, for and , using Hölder inequality, Proposition 14, and the boundedness of the fractional integral , we obtain, for any , where we have chosen large enough.

Secondly, for any , where we have also chosen large enough.

Therefore,

Therefore, if , then

Thus, we have Note that, for , where .

By Theorem 4, we get Therefore,

This completes the proof of Theorem 5.

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