Research Article | Open Access

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

**Academic Editor:**Chuanzhi Bai

#### 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

- G. Rozenblum and G. Tashchiyan, “Riesz ${L}^{p}$ summability of spectral expansions related to the Schrödinger operator with constant magnetic field,”
*Journal of Mathematical Analysis and Applications*, vol. 284, no. 1, pp. 315–331, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. Carleson and P. Sjölin, “Oscillatory integrals and a multiplier problem for the disc,”
*Studia Mathematica*, vol. 44, pp. 287–299, 1972. View at: Google Scholar | Zentralblatt MATH | MathSciNet - A. Cordoba, “The Kakeya maximal function and the spherical summation multipliers,”
*American Journal of Mathematics*, vol. 99, no. 1, pp. 1–22, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Fefferman, “A note on spherical summation multipliers,”
*Israel Journal of Mathematics*, vol. 15, pp. 44–52, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. A. Tomas, “A restriction theorem for the Fourier transform,”
*Bulletin of the American Mathematical Society*, vol. 81, pp. 477–478, 1975. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Carbery, “The boundedness of the maximal Bochner-Riesz operator on ${L}^{4}({\mathbb{R}}^{2})$,”
*Duke Mathematical Journal*, vol. 50, no. 2, pp. 409–416, 1983. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Christ, “On almost everywhere convergence of Bochner-Riesz means in higher dimensions,”
*Proceedings of the American Mathematical Society*, vol. 95, no. 1, pp. 16–20, 1985. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Fefferman, “The multiplier problem for the ball,”
*Annals of Mathematics*, vol. 94, pp. 330–336, 1971. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Carbery, J. L. Rubio de Francia, and L. Vega, “Almost everywhere summability of Fourier integrals,”
*Journal of the London Mathematical Society*, vol. 38, no. 3, pp. 513–524, 1988. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. Garcia-Cuerva and J. L. Rubio De Francia,
*Weighted Norm Inequalities and Related Topics*, North-Holland, Amsterdam, The Netherlands, 1985.

#### Copyright

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.