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International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 214030, 8 pages
http://dx.doi.org/10.1155/2008/214030
Research Article

Some Estimates of Schrödinger-Type Operators with Certain Nonnegative Potentials

1Department of Mathematics and Mechanics, University of Science and Technology Beijing, Beijing 100083, China
2College of Science, Shandong Jianzhu University, Jinan 250101, Shandong Province, China

Received 4 September 2008; Revised 1 November 2008; Accepted 3 November 2008

Academic Editor: Jie Xiao

Copyright © 2008 Yu Liu and Youzheng Ding. 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

We consider the Schrödinger-type operator , where the nonnegative potential belongs to the reverse Hölder class for ,  . The estimates of the operator related to are obtained when and . We also obtain the weak-type estimates of the operator under the same condition of .

1. Introduction

In recent years, there has been considerable activity in the study of Schrödinger operators (see [14]). In this paper, we consider the Schrödinger-type operator where the potential belongs to for . We are interested in the boundedness of the operator , where the potential satisfies weaker condition than that in [5, Theorem 1, (2)]. The estimates of some other operators related to Schrödinger-type operators can be found in [2, 5].

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 .

It follows from [3] that the class has a property of “self-improvement”, that is, if , then for some .

We now give the main results for the operator in this paper.

Theorem 1.1. Suppose Then for there exists a positive constant such that

By the proof of Theorem 1.1, we obtain the following weak-type estimate.

Theorem 1.2. Suppose , Then for there exists a positive constant such that

Under a stronger condition on the potential , Sugano [5] has obtained the following proposition.

Proposition 1.3. Suppose and there exists a constant such that Then for there exists a positive constant such that

As a direct consequence of our estimates, we have the following corollary.

Corollary 1.4. Suppose for Assume that in Then

Throughout this paper, unless otherwise indicated, we will use to denote constants, which are not necessarily the same at each occurrence. By , we mean that there exist constants and such that .

2. The Auxiliary Function and Estimates of Fundamental Solution

In this section, we firstly recall the definition of the auxiliary function and some lemmas about the auxiliary function which have been proven in [3].

Lemma 2.1. If , then the measure satisfies the doubling condition, that is, there exists such that holds for all balls in

Lemma 2.2. For and for there exists such that

Assume that , . The auxiliary function is defined by

Lemma 2.3. If , then Moreover,

Lemma 2.4. There exists such that for any and in In particular, if

Lemma 2.5. There exists such that

Lemma 2.6. There exists and such that, for any

Refer to [3] for the proof of the above lemmas.

The next lemma has been obtained by Tao and Wang in [6].

Lemma 2.7. Let and be sufficiently large, then there are positive constants and such that for any and

In order to prove Theorem 1.1, we need to give the estimates of the fundamental solution of . Zhong has established the estimates of the fundamental solution of in [2] when is a nonnegative polynomial. Recently, Sugano [5] has obtained the polynomial decay estimates of the fundamental solution of under a weaker condition on in the following theorem.

Theorem 2.8. Assume and let be the fundamental solution of For any positive integer there exists a constant such that

3. Proof of the Main Results

In this section, we will prove Theorems 1.1 and 1.2.

Theorem 3.1. Suppose Then for there exists a positive constant such that

Proof. Let and We need to show that Write where .
Because of the self-improvement of the class, for some , we have where .
Thus, Now, let . Then where we used (1.2), Lemmas 2.3 and 2.4.
Hence, we have proved that for some , By choosing , and in Lemma 2.7, we immediately have Thus, Therefore, by using interpolation we have Then we deal with .
For , by the Hölder inequality, where and we apply the second inequality for and in Lemma 2.7 to the last step.
Thus, for , Fix and let . By Lemmas 2.4, 2.6, and 2.7, if we choose large enough.
From this, we have Thus the theorem is proved.

Now we give the proof of Theorem 1.1.

Proof of Theorem 1.1. Suppose for some . By Theorem 3.1, we have It follows that Because is a Calderón-Zygmund operator, for , we have

Proof of Theorem 1.2. Note that satisfies
Thus, by the proof of Theorem 1.1,

Acknowledgments

I would like to express my gratitude to the referee for the valuable comments. This work is supported by the Tian Yuan Project of the National Natural Science Foundation (NNSF) of China under Grant no. 10726064.

References

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