Research Article | Open Access

Liang Song, Chaoqiang Tan, "Hardy Spaces Associated to Schrödinger Operators on Product Spaces", *Journal of Function Spaces*, vol. 2012, Article ID 179015, 17 pages, 2012. https://doi.org/10.1155/2012/179015

# Hardy Spaces Associated to Schrödinger Operators on Product Spaces

**Academic Editor:**Hans G. Feichtinger

#### Abstract

Let be a Schrödinger operator on , where is a nonnegative function on . In this article, we show that the Hardy spaces on product spaces can be characterized in terms of the Lusin area integral, atomic decomposition, and maximal functions.

#### 1. Introduction

Let be a nonnegative function on . The Schrödinger operator with potential is defined by The operator is a self-adjoint positive definite operator. From the Feynman-Kac formula, it is well known (see, e.g., [1, page 195]) that the kernel of the semigroup satisfies the estimate for all and .

Let us consider the Hardy space on product domains. We note that the usual Hardy space on the product domain is now well understood (see, e.g., [2–4]). In this paper we will be concerned with the space associated to as introduced in [5] (see [6, 7] for one-parameter theory). Firstly, we set and note that where (resp., ) stands for the range (resp., the nullspace) of , and the sum is orthogonal. For a function , define where The space is defined as the completion of in the norm given by

The main purpose of this article is to derive atomic characterizations and the maximal characterizations of . Before stating our results, let us recall some necessary notations (see also [8, 9]). Suppose that is an open set with finite measure. Denote by the maximal dyadic subrectangles of in the form of , where and are cubes in . Let and we denote by and the side lengths of and , respectively. For given , we will write for the -fold dilation of with the same center.

An -atom is a function on , together with an open set of finite measure, which satisfies the following properties:(i) (see (2.4));(ii) can be further decomposed as , where (iii).

We can define the atomic Hardy space by with the norm given by the natural quotient norm:

The first result of this paper is the following theorem.

Theorem 1.1. *Let be the Schrödinger operator as (1.1). Then the spaces and coincide. In other words,
*

Next we give the “maximal" characterizations of . Given a function we define two functions: Similarly, one can consider the Poisson semigroup generated by the operator and the operators with and .

Define the spaces , , and as the completion of in the norms given by the norm of the corresponding square or maximal functions, respectively. For example, By a similar manner, the norms of , and are defined. The second result of this paper is the following.

Theorem 1.2. *Let be the Schrödinger operator as (1.1). Then the spaces , , , and coincide with equivalent norms.*

This paper is organized as follows. In Section 2, we will give some preliminary results including the properties of Schrödinger operators and tent spaces on product spaces. The proofs of Theorems 1.1 and 1.2 will be given in Sections 3 and 4, respectively.

Throughout this paper, the letter “" or “" will denote (possibly different) constants that are independent of the essential variables.

#### 2. Preliminaries

##### 2.1. Tent Spaces on Product Domains

A theory of “tent spaces" was developed by Coifman et al. [10, 11]. These spaces are useful for the study of a variety of problems in harmonic analysis. In particular, we note that the tent spaces give a natural and simple approach to the atomic decomposition of functions in the classical Hardy space by using the area integral functions and the theory of the Carleson measure. See also [6, 12].

Tent spaces have been studied by [13, 14] in connection with the theory of Carleson measures on product domains. Let be the usual upper half-space in . For any , we set as the standard cone (of aperture ) with vertex . In particular, we set . If , then denotes the rectangle centered at whose side lengths are and , respectively. For any open set , the tent over , , is the set For any function defined on we will write The tent space is then defined as the space of functions such that and is equipped with the norm, , .

We now introduce -atoms.

*Definition 2.1. *A function is called a -atom if there exists an open set of finite measure satisfying the following properties:(i) can be further decomposed as , where each is supported in , and is a maximal dyadic subrectangle of in the form of , where and are cubes in ;(ii) and .

Proposition 2.2. *Suppose . Then , where are -atoms, , and so that the sum converges in the norm. Moreover, if one assumes that , then the sum also converges in the norm.*

* Proof. *See [5, Proposition 3.3] for the proof.

##### 2.2. Some Results on Product Spaces

We recall that the strong maximal function is defined as follows: where and are cubes in . It is well known that the operator is bounded on , for .

Now for any open set with finite measure, we set By the strong maximal theorem, . Denote by the dyadic subrectangles that are maximal in the direction, where are dyadic cubes in . Define similarly. It is well known that Journé’s covering lemma holds (see [8, 15]).

Lemma 2.3. *Let . For any , one sets . Define similarly. Then for any , one has
**
where is a constant depending only on , but not on .*

The following lemma shows that in order to prove that an operator is bounded from to , we just need to check that the operator is uniformly bounded on the -atoms.

Lemma 2.4. *Assume that is either a linear operator or a positive sublinear operator, bounded on and for every -atom ,
**
with constant independent on . Then can extend to a bounded operator from to , and
*

* Proof. *Its proof is similar to that of [16, Lemma 3.3] and we omit it here. See also [17].

##### 2.3. Some Properties of the Schrödinger Operator on

Let be the Schrödinger operator as (1.1), and let be the kernels of the operators of semigroup .

First we note that, for each , there exist two positive constants and such that the time derivatives of satisfy for all and almost all . For the proof, see, for example, [18, 19].

Next, for , we define Then for any nonzero function , we have that . Denote . It follows from the spectral theory in [20] that for any , where . As an application, we have for some constant .

Lemma 2.5. *Let and . Then for any , there exists a constant such that
**
where with and .*

* Proof. *For the proof, we refer to [7, Lemma 8.4].

Recall that if is a nonnegative, self-adjoint operator on , and denotes its spectral decomposition, then for every bounded Borel function , one defines the operator by the formula In particular, the operator is then well defined on . Moreover, it follows from [21, Theorem 3] that the Schwartz kernel of satisfies See also [22]. By the Fourier inversion formula, whenever is an even bounded Borel function with the Fourier transform of , , we can write in terms of . In fact, using (2.12) we have which, when combined with (2.13), gives

Lemma 2.6. *Let be even, , , and set and for . Let and denote the Fourier transform of and ,respectively. Then, the kernels of the operators and have supports contained in .*

* Proof. *For the proof, we refer the reader to [7, Lemma 3.5].

Lemma 2.7. *Let as in Lemma 2.6; then the operator
**
is bounded from to , if .*

* Proof. *For the proof, we refer to [5, Lemma 3.4].

#### 3. Proof of Theorem 1.1

##### 3.1. The Inclusion of

Let . Then . We start with a suitable version of the Calderón reproducing formula. Let be as in Lemma 2.6, , and let be a constant such that . By -functional calculus [23], one can write where the integral converges in . By Proposition 2.2, has a -atomic decomposition: , where , and are -atoms associated to an open set . It is easy to see that the sum converges in and . We have where and . Here is a large constant determined later. Using Lemma 2.7, we can show that the sum (3.2) converges in .

It is known that . Now we turn to check that are -atoms associated to open sets . Since are -atoms, then . Thus Lemma 2.6 implies that for , and .

To continue, we write if we choose large enough. By a similar argument, we have

For general , we just need a standard argument. The inclusion of is finished.

##### 3.2. The Inclusion of

By Lemma 2.4, it is enough to show that for any -atom associated to an open set . For , we let be the maximal dyadic cube such that and . Also we let be the maximal dyadic cube such that and . Let . We can see that , since
Due to Hölders inequality one has
Let us prove . One can write
We only estimate the term since the proof of the term is similar. Observe that
Consider term I_{1}. By Hölder’s inequality, we obtain
Consider the term I_{11}. It follows from estimate (2.8) that
Let denote the center of cube . Note that and . We use Hölder’s inequality to obtain
where in the last inequality we use Lemma 2.3.

For the term I_{12}, we apply the definition of -atom to obtain
which, together with estimate of I_{11}, show that . By a similar argument as mentioned previously, we can show . We have obtained the required estimate . This completes the proof of Theorem 1.1.

#### 4. Proof of Theorem 1.2

In this section, we will give the proof of Theorem 1.2 in the following routine: . The main idea comes from [9, 16].

*Step I*. . It is similar to the proof of . Here we omit the details.

*Step II*. ) because .

*Step III*. . By -functional calculus ([23]), we have
Therefore,
which completes the proof of .

*Step IV*. , for any , and . Applying Lemma 2.5 with , we obtain
Therefore,
This completes the proof of .

Lemma 4.1. *Let and be the functions of , and suppose that is radial and supp. Let . Then one has
**
where with is a vector-value function independent on and and .*

* Proof. *We note that
Following the steps in [24], we obtain
Then the lemma follows readily.

Applying Lemma 4.1, we can obtain the following lemma.

Lemma 4.2. *Suppose that . Let and be functions as in Lemma 4.1. Then one has
*

* Proof. *The proof of Lemma 4.2 can be obtained by iterating Lemma 4.1.

We begin to show by following the idea of [9] (see also [25, pages 107–109]).

Set and .

We claim that Once the claim holds, we integrate from 0 to to complete the proof of . Now we turn to prove the claim.

The boundedness of the strong maximal operator on implies To prove (4.9), one just need to estimate Observe that where and . Now we choose a radial nonnegative function such that if and if . Also set . We can easily check that if , then , for some positive constant . Applying Lemma 4.2, we have

We firstly consider the term I. If , then , for some . As a consequence, . Thus Consider the term II. If , then there exists , such that and . Thus, . We obtain Similarly, . For the term , it follows from the fact that

Summarizing the estimates aforementioned, we have which completes the proof of .

*Step V*. . It is similar to the proof of . We omit the details.

#### Acknowledgments

The authors thank Y. S. Han and L. X. Yan for helpful suggestions. L. Song is supported by NNSF of China (no. 11001276) and the Fundamental Research Funds for the Central Universities (no. 11lgpy79). C. Q. Tan is supported by Specialized Research Fund for the Doctoral Program of Higher Education (no. 20104402120002) and NSF of Guangdong (no. 10451503101006384).

#### References

- E. M. Ouhabaz,
*Analysis of Heat Equations on Domains*, vol. 31 of*London Mathematical Society Monographs*, Princeton University Press, Princeton, NJ, USA, 2005. - S.-Y. A. Chang and R. Fefferman, “A continuous version of duality of ${H}^{1}$ with BMO on the bidisc,”
*Annals of Mathematics*, vol. 112, no. 1, pp. 179–201, 1980. View at: Publisher Site | Google Scholar - S.-Y. A. Chang and R. Fefferman, “The Calderón-Zygmund decomposition on product domains,”
*American Journal of Mathematics*, vol. 104, no. 3, pp. 455–468, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Fefferman, “Calderón-Zygmund theory for product domains: ${H}^{p}$ spaces,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 83, no. 4, pp. 840–843, 1986. View at: Publisher Site | Google Scholar - D. G. Deng, L. Song, C. Tan, and L. X. Yan, “Duality of Hardy and BMO spaces associated with operators with heat kernel bounds on product domains,”
*The Journal of Geometric Analysis*, vol. 17, no. 3, pp. 455–483, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - P. Auscher, X. T. Duong, and A McIntosh, “Boundedness of Banach space valued singular integral operators and Hardy spaces,” Unpublished preprint. View at: Google Scholar
- S. Hofmann, G. Z. Lu, D. Mitrea, M. Mitrea, and L. X. Yan, “Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaeney estimates,”
*Memoirs of the American Mathematical Society*, vol. 214, no. 1007, 2011. View at: Google Scholar - R. Fefferman, “Harmonic analysis on product spaces,”
*Annals of Mathematics*, vol. 126, no. 1, pp. 109–130, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH - R. Gundy and E. M. Stein, “${H}^{p}$ theory for the poly-disc,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 76, no. 3, pp. 1026–1029, 1979. View at: Publisher Site | Google Scholar - R. R. Coifman, Y. Meyer, and E. M. Stein, “Un nouvel espace adapté a l'étude des opérateurs déefinis par des intégrales singuliéres.,”
*Lecture Notes in Mathematics*, vol. 992, pp. 1–15, 1983. View at: Google Scholar - R. R. Coifman, Y. Meyer, and E. M. Stein, “Some new function spaces and their applications to harmonic analysis,”
*Journal of Functional Analysis*, vol. 62, no. 2, pp. 304–335, 1985. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. T. Duong and L. X. Yan, “Duality of Hardy and BMO spaces associated with operators with heat kernel bounds,”
*Journal of the American Mathematical Society*, vol. 18, no. 4, pp. 943–973, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Alvarez and M. Milman, “Spaces of Carleson measures: duality and interpolation,”
*Arkiv för Matematik*, vol. 25, no. 2, pp. 155–174, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Gomez and M. Milman, “Complex interpolation of ${H}^{p}$ spaces on product domains,”
*Annali di Matematica Pura ed Applicata*, vol. 155, pp. 103–115, 1989. View at: Publisher Site | Google Scholar - J. L. Journé, “A covering lemma for product spaces,”
*Proceedings of the American Mathematical Society*, vol. 96, no. 4, pp. 593–598, 1986. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Hofmann and S. Mayboroda, “Hardy and BMO spaces associated to divergence form elliptic operators,”
*Mathematische Annalen*, vol. 344, no. 1, pp. 37–116, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. Han, G. Lu, and K. Zhao, “Discrete Calderón's identity, atomic decomposition and boundedness criterion of operators on multiparameter Hardy spaces,”
*Journal of Geometric Analysis*, vol. 20, no. 3, pp. 670–689, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. T. Duong and D. W. Robinson, “Semigroup kernels, Poisson bounds, and holomorphic functional calculus,”
*Journal of Functional Analysis*, vol. 142, no. 1, pp. 89–128, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. Saloff-Coste, “Analyse sur les groupes de Lie á croissance polynómiale,”
*Arkiv för Matematik*, vol. 28, no. 2, pp. 315–331, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH - E. B. Davies,
*Heat Kernels and Spectral Theory*, Cambridge University Press, Cambridge, Mass, USA, 1989. View at: Publisher Site - T. Coulhon and A. Sikora, “Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem,”
*Proceedings of the London Mathematical Society*, vol. 96, no. 2, pp. 507–544, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Cheeger, M. Gromov, and M. Taylor, “Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds,”
*Journal of Differential Geometry*, vol. 17, no. 1, pp. 15–53, 1982. View at: Google Scholar | Zentralblatt MATH - A. McIntosh, “Operators which have an ${H}_{\infty}$-calculus. Miniconference on operator theory and partial dierential equations,” in
*The Centre for Mathematics and its Applications (ANU '86)*, vol. 14, pp. 210–231, Canberra, Australia, 1986. View at: Google Scholar - K. Merryfield, “On the area integral, Carleson measures and ${H}^{p}$ in the polydisc,”
*Indiana University Mathematics Journal*, vol. 34, no. 3, pp. 663–685, 1985. View at: Publisher Site | Google Scholar - E. M. Stein,
*Beijing Lectures in Harmonic Analysis*, Princeton University Press, Princeton, NJ, USA, 1986.

#### Copyright

Copyright © 2012 Liang Song and Chaoqiang Tan. 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.