Hardy Spaces Associated to Schrödinger Operators on Product Spaces
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.
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  (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.
2.1. 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
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 .
Next, for , we define Then for any nonzero function , we have that . Denote . It follows from the spectral theory in  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 . 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 , 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 I1. By Hölder’s inequality, we obtain Consider the term I11. 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 I12, we apply the definition of -atom to obtain which, together with estimate of I11, 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
Step I. . It is similar to the proof of . Here we omit the details.
Step II. ) because .
Step III. . By -functional calculus (), 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 , 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
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.
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).
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