Abstract and Applied Analysis

Volume 2013 (2013), Article ID 859680, 7 pages

http://dx.doi.org/10.1155/2013/859680

Research Article

Almost Everywhere Convergence of Riesz Means Related to Schrödinger Operator with Constant Magnetic Fields

Liurui Deng¹ and Bolin Ma²

¹College of Finance and Statistics, Hunan University, Changsha 410082, China

²College of Science and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China

Received 25 September 2012; Accepted 28 January 2013

Academic Editor: Chuanzhi Bai

Copyright © 2013 Liurui Deng and Bolin Ma. 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 study almost everywhere convergence for Riesz means related to Schrödinger operator with constant magnetic fields. Through researching the weighted norm estimates for the maximal operator with power-weight functions, we obtain the desired result, which is similar to the work given by Anthony Carbery, Jose L. Rubio de Francia, and Luis Vega.

1. Introduction

The magnetic Schrödinger operator (MSO) with constant magnetic fields in is of the form

where is a real antisymmetric matrix. If is degenerated (this requires to be odd, and is the rank of the matrix ), then its eigenvalues have the form , . In properly chosen coordinates , the operator becomes

where is the Laplacian in . The spectrum of is in the semiaxis starting from the point , and its spectral expansion is continuous (see [1]).

Let ’s be positive. Denote by the spectral function of . It is an integral operator with a kernel , which is skew-translation invariant; that is, .

For , set the Riesz means of order as

and the kernel of as

with the same skew-translation invariance. We will denote by the corresponding maximal operator; that is,

As an indispensable part in harmonic analysis, many people investigate the convergence of Bochner-Riesz means for Fourier transform in norm and almost everywhere, which is defined as

Since the convergence of in -norm is equivalent to the boundedness of in , persons look for the boundedness of it. For and , Carleson, Cordoba, and Fefferman turn out the boundedness in (see [2–4]). When , it is only proven for (see [4, 5]). Returning the problem about almost everywhere convergence of , Carbery has finished it for and in two dimensions in 1983 (see [6]). For higher dimensions, it is completed by Christ only for (see [7]). In the special case, , Fefferman studies the -boundeness of for (see [8]). Not until 1988 was it solved by Carbery et al. for , , and (see [9]).

In [1], Rozenblum and Tashchiyan investigated the -norm convergence for Riesz means for Schrödinger operator with constant magnetic fields. They showed that under the restriction theorem similar to one for Fourier transform in [5], it is of the same results as Bochner-Riesz means in . However, very few results are considered for almost everywhere convergence of Riesz means for Schrödinger operator with constant magnetic fields. In the paper, we are interested in it. Through researching the boundedness of the maximal operator in , we get the desired result.

2. Main Results

Theorem 1. Set and . Write . If with , then almost everywhere.

Usually, we replace the almost everywhere convergence of Riesz means with estimates for the maximal operator. However, we only need to think about weighted estimates for the maximal operator , as follows. In fact, based on the idea in [9], for , there exists a number with such that . We have gotten boundedness of the maximal operator in another paper.

Theorem 2. Suppose that and . Then,

In order to prove the theorem, we introduce the following lemmas, which are the essential tools. In the following lemmas, we reduce the maximal operator to the one generated by a multiplier with compact support. Since it is easy to see that the boundedness of the maximal operator generated by the multipliers is independent of the index , the dimension will not play an important role in the following estimates.

Lemma 3 (see [9]). If  , then

where

For a small , let be a function supported in and satisfy

Define

Let supported in and for . Set

Denoting by the character function of the set , write

Let be a function with Fourier transform as

Set

Accordingly, define the operator as

where is the Fourier transform of . Apparently, since

we decompose

Lemma 4. For , one has

where the constant is independent of and .

Proof. With the method similar to the proof of Lemma  4 in [9], we write and expand into a Taylor series around . Then,

where the remainder satisfies

But is a Schwartz function and can be integrated against . Hence,

Since is a resolution of the identity, we see

Denote by the kernel of . For almost all , we let . Decompose

We have

It is easy to show

Lemma 5. If  and , then

where is independent of and .

Proof. Suppose that is supported in . Write . If , then . Since

we only need to prove that

For the case of , with Lemma 4, it follows that

It is easy to see that

Thus, we have

Choosing , we get

On the other hand, is self-adjoint. So,

Hence,

With

we get

and it implies that is also self-adjoint; that is,

Therefore, by duality,

Taking , we can establish the inequality

Nextly, we consider . By the definition of and , we see that the kernel of is supported in

and is supported in

So, the support of is contained in

With Lemma 4, we have

where .

At last, we turn to the case of . Similar to the aforementioned, we have

By duality again, we see

Through we choose , it is not difficult to get

Combining (28), (40),(44) with (47), when is supported in , we come to the conclusion

Now, we hope to establish (48) for all . Decompose , where are disjoint cubes of common side with centered at . Since have essentially disjoint supports, it suffices to prove (48) for every . When , we have proved it. If , then

Therefore, we only need to confirm

In fact, it follows from Lemma 4 that

At present, we complete the proof of Lemma 5.

Lemma 6. For , , and , one gets

Proof. Applying Minkowski and Cauchy-Schwartz’s inequalities into the left hand side of (52), we get

Now, it suffices to prove that

where is uniform in . For , it is equivalent to

It is just as the result of Lemma 5.

Now, we come back to the proof of Theorem 2.

Proof. As in [3], we can decompose

Thus,

Consequently, we consider

Let

and be defined in the same way but using instead of the function

which satisfies the same estimates as . Then, by the fundamental theorem in calculus and Hölder’s inequality, we have

Take a Schwartz function such that and if and . Then, when , we have

Using Lemma 6, we get

From Theorem 3.1 in page 411 of [10] and the density of in , we induce that

is bounded in . As a result,

At last, with (61) and Hölder’s inequality, we come to the result that

Acknowledgments

The authors deeply thank the referee for reviewing their paper. The first author was supported by the NSF of China (71201051, 71031004, and 71073047), NSF of Hunan Province (2012RS4028) and Postdoctoral Science Foundation of China (2012M521513). The second author was supported by the NSF of China (10771054) and NSF of Hunan Province (09JJ5002).

References

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