Boundedness of Singular Integrals on Hardy Type Spaces Associated with Schrödinger Operators

Abstract

Let be a Schrödinger operator on , where is a nonnegative potential belonging to the reverse Hölder class . The Hardy type spaces for some , are defined in terms of the maximal function with respect to the semigroup . In this paper, we investigate the bounded properties of some singular integral operators related to , such as and , on spaces . We give the molecular characterization of , which is used to establish the -boundedness of singular integrals.

1. Introduction

Let be a Schrödinger operator on , where is a nonnegative potential belonging to the reverse Hölder class for some ; that is, there exists a constant such that the reverse Hölder inequality holds for every ball in . It is well known that if then for some . Also obviously, when .

Some singular integral operators related to , such as the imaginary power , and the Riesz transform have been studied by Shen [1]. Some of his results are following. The operator is a Calderón-Zygmund operator for any . is a Calderón-Zygmund operator if . When , is bounded on for , where . The above range of is optimal. Earlier results were given by Fefferman [2] and Zhong [3].

The Hardy type spaces for some , associated with , are studied by Dziubański and Zienkiewicz [4, 5]. They establish the atomic decomposition theorem and the Riesz transform characterization of . Specifically, is bounded from to . We will investigate the bounded properties of the operators and on spaces . To do this, we give the molecular characterization of .

Without loss of generalization, we assume that for some and set . When , we set . Throughout the paper, we will use and to denote the positive constants, which are independent of main parameters and may be different at each occurrence. By , we mean that there exists a constant such that .

Let be the semigroup of linear operators generated by and their kernels. Since is nonnegative, the Feynman-Kac formula implies that where is the convolution kernels of the heat semigroup . The estimate (2) can be improved as follows. We introduce the auxiliary function defined by It is known that . For every , (cf. [6, Theorem 4.10]). Let ; for every   and all , (cf. [6, Proposition 4.11]).

We define the Hardy type spaces , in terms of the maximal function with respect to the semigroup .

For , the Hardy space is defined, according to Dziubański and Zienkiewicz [4], by where The norm of a function is defined to be .

The Hardy spaces, , consist of some kind of distributions. But may have no meaning for a tempered distribution because are not smooth. Let be a locally integrable function. is the ball of radius centered at . Set Let . A locally integrable function is said to be in the Campanato type space if All spaces are mutually coincident with equivalent norms and will be simply denoted by (cf. [7]). Due to (4) and (5), for every , (cf. [7, Lemma 1]). Thus the semigroup maximal function is well defined for distributions in . We define the Hardy space, , by and set .

Similar to the classical case, the Hardy space admits an atomic decomposition. Let . A function is called an -atom associated with a ball if(1),(2),(3).

Proposition 1 (see [7, Theorem 1]). Given as above, then if and only if can be written as , where are -atoms and . The sum converges in norm and also in when . Moreover, where the infimum is taken over all decompositions of into -atoms.

Now we state the main results in this paper.

Theorem 2. For any , the imaginary power is bounded on for . When , the Riesz transform is bounded on for . Moreover, is bounded on whenever .

Remark 3. When , the kernel of Riesz transform only satisfies the Hörmander condition with respect to the second variable, which is weaker than the smoothness condition of standard kernels. Thus we cannot expect, in general consideration, to deal with the boundedness of for the case of .

In order to prove Theorem 2, we give the molecular characterization of .

Let and . Set . A function is called an -molecule with the center if(1),(2),(3), where is the volume of the unit ball.

Theorem 4. Given as above, then if and only if can be written as , where are -molecules and . The sum converges in norm and also in when , where are -molecules. Moreover, where the infimum is taken over all decompositions of into -molecules.

Remark 5. It is easy to verify that any -atom is an -molecule with a constant factor less than or equal to . We will see that the image of an -atom under the action of a singular integral operator may not be an -molecule but is a sum of two -molecules up to constant factors. This is different from the classical case.

This paper is organized as follows. In Section 2, we collect some useful facts and results about the potential , the auxiliary function and the kernels of operators , and , which will be used in the sequel. Most of these results are already known. In Section 3, we prove Theorem 4. The proof of Theorem 2 is given in the last two sections. The -boundedness for is proved in Section 4 while -boundedness is proved in Section 5.

2. Preliminaries

First we list some known facts and results about the potential , the auxiliary function , and the kernels of operators and .

Lemma 6. is a doubling measure; that is, there exists a constant such that

Lemma 7. Consider

Lemma 8. There exists such that

Lemma 9. There exists such that In particular, if .

Let and be the kernels of and ,  respectively, and   and the kernels of and , respectively. Set .

Lemma 10. is a Calderón-Zygmund operator. It does not matter to assume that . The kernel satisfies and, for any , In addition,

Lemma 11. When , is bounded on for , where . The kernel satisfies, for any , In addition,

Lemma 12. When , is a Calderón-Zygmund operator. The kernel satisfies and, for any , In addition, for any ,

For Lemmas 6–12, we refer readers to [1]. We also need the following estimates about and .

Lemma 13. When , When , for any ,

Proof. It is well known that Therefore, we also have the estimates We may assume that . Otherwise, Lemma 13 is obvious.
We will use the following known facts (cf. [1]). Let and denote, respectively, the fundamental solutions for the operators and in , where . They satisfy the following estimates. For any and , when . Set . Then is expressed as Thus, Note that for some . Using Hölder inequality and condition, it is easy to see that, for , Note that when . Making use of (34), we get where we have used Lemma 7 in the last inequality. Similarly, To estimate , we write Using Hölder inequality and condition, we obtain Using Lemma 6 and taking sufficiently large, we get Therefore, We also have where satisfies the estimate If , by the same argument as (40), for any , By the functional calculus and making use of (40), we obtain This proves (26).
Similarly, it follows from (43) that This proves (27).

3. Molecular Characterization

Essentially, the proof of Theorem 4 is the same as the usual molecular theory.

Proof of Theorem 4. By Proposition 1, it is sufficient to prove that for any-molecule admits an atomic decomposition , where are -atoms and .
We will give the proof in case . The proof is similar in the case of . Suppose is an -molecule centered at . Let , where . If , we return the usual molecular theory (cf. [8]). Thus nothing needs to be proved. Suppose . Set Then Note that and . Also we have Thus , where are -atoms and .
Originally, the sum in (47) converges pointwise. When , it is easy to see that the sum in (47) converges in . If , for any , Therefore, It follows that the sum in (47) converges in . The proof of Theorem 4 is completed.

4. -Boundedness

In this section, we prove the boundedness of on . When , the boundedness of on , can be proved by the same method. In fact, their kernels satisfy similar estimates.

Let be an -atom associated with a ball for some suitable . If , we will prove that is an -molecule up to a constant factor. If , may be not an -molecule up to a constant factor but is (cf. [9]). We will prove that is an -molecule up to a constant factor for some suitable . This means that uniformly. Because the semigroup maximal function is subadditive, by Proposition 1, is bounded on .

First, let . Because where , we have Thus there needs no the cancelation condition. We only need to estimate . Write It is obvious that For , if , by Lemma 9, . Note that when . Using Lemma 10, we get and provide  . Therefore, It follows that

Next, suppose . Let us estimate . Consider Note that , when and by Lemma 10, we have Here we choose such that and or, equivalently, . When , using the cancelation condition of and Lemma 13, we obtain It follows that When , by (29), we have Then We have seen that As above, there needs no the cancelation condition. To finish the proof, we only need to prove or, equivalently, Write It is clear that By (63), where we have taken such that , which implies that . The proof is complete.

5. -Boundedness

In this section we prove the boundedness of on when .

Let be an -atom associated with a ball for some suitable . As the above section, if , we will prove that is an -molecule up to a constant factor. If , we will prove that is an -molecule up to a constant factor for some suitable . In any case we have uniformly.

Suppose . It follows from Lemma 11 that provide   , where . Thus there needs no the cancelation condition. Write It is obvious that On the other hand, we have Note that when and and by Lemma 11, Since for , it is clear that provide  . We have taken such that , where . Let . Then . Using the theorem on fractional integrals, condition, and Lemma 8, we obtain provide   sufficiently large. Thus . It follows that Therefore, and .

In case , we need to prove that is an -molecule up to a constant factor for some suitable . First we give the estimate of . Write We have By Lemma 11, Note that when . It is obvious that provide  . On the other hand, using the theorem on fractional integrals and condition with , and , we get where we have used Lemma 7 in the last second inequality. Thus, Since when and , , it is easy to see that Therefore we have As above, there needs no the cancelation condition. Write It is obvious that We have Since and when and , by Lemma 11, Similar to in the proof of (77), we obtain by the same argument. It follows that Using the cancelation condition of , where we have taken such that . This proves that It follows that . The proof is completed.

Conflict of Interests

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

Acknowledgments

This work was supported by Research Fund for the Doctoral Program of Higher Education of China (20113108120001), the National Natural Science Foundation of China under Grants nos. 11471018 and 11371036, and the Beijing Natural Science Foundation under Grant no. 1142005.