Hardy Spaces Associated to Schrödinger Operators on Product Spaces

Let L = − Δ + V be a Schrodinger operator on ℝ n , where V ∈ L l o c 1 ( ℝ n ) is a nonnegative function on ℝ n . In this article, we show that the Hardy spaces L on product spaces can be characterized in terms of the Lusin area integral, atomic decomposition, and maximal functions.


Introduction
Let V ∈ L 1 loc R n be a nonnegative function on R n .The Schr ödinger operator with potential V is defined by The operator L 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 h t x, y of the semigroup e −tL satisfies the estimate 0 ≤ h t x, y ≤ 1 4πt n/2 e −|x−y| 2 /4t 1.2 for all t > 0 and x, y ∈ R n .Let us consider the Hardy space on product domains.We note that the usual Hardy space H 1 R n × R n on the product domain is now well understood see, e.g., 2-4 .In this 2 Journal of Function Spaces and Applications paper we will be concerned with the space H 1 L R n × R n associated to L as introduced in 5 see 6, 7 for one-parameter theory .Firstly, we set and note that where R L ⊗ L resp., N L ⊗ L stands for the range resp., the nullspace of L ⊗ L, and the sum is orthogonal.For a function f ∈ L 2 R n × R n , define where 1.6 The space H 1 L R n × R n is defined as the completion of H 2 R n × R n in the norm given by The main purpose of this article is to derive atomic characterizations and the maximal characterizations of H 1 L R n × R n .Before stating our results, let us recall some necessary notations see also 8,9 .Suppose that Ω ⊂ R n × R n is an open set with finite measure.Denote by m Ω the maximal dyadic subrectangles of Ω in the form of R I × J, where I and J are cubes in R n .Let R I × J ∈ m Ω and we denote by I and J the side lengths of I and J, respectively.For given λ > 0, we will write λR for the λ-fold dilation of R I × J with the same center.
An L-atom is a function a on R 2n , together with an open set Ω of finite measure, which satisfies the following properties: ii a x can be further decomposed as a R∈m Ω a R , where We can define the atomic Hardy space with the norm f H 1 atom R n ×R n given by the natural quotient norm: inf The first result of this paper is the following theorem.
Theorem 1.1.Let L be the Schrödinger operator as 1.1 .Then the spaces Next we give the "maximal" characterizations of

1.12
Similarly, one can consider the Poisson semigroup generated by the operator L and the operators the norms given by the L 1 norm of the corresponding square or maximal functions, respectively.For example, By a similar manner, the norms of The second result of this paper is the following.
Theorem 1.2.Let L be the Schrödinger operator as 1.1 .Then the spaces This paper is organized as follows.In Section 2, we will give some preliminary results including the properties of Schr ödinger 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 "C" or "c" will denote possibly different constants that are independent of the essential variables.

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 H p R n 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 R n 1 be the usual upper half-space in R n 1 .For any α > 0, we set For any function f y, t defined on R n 1 × R n 1 we will write

2.2
The tent space T p 2 is then defined as the space of functions f such that Af ∈ L p R 2n and is equipped with the norm, f T p finite measure satisfying the following properties: i A x, t can be further decomposed as A R∈m Ω A R , where each A R is supported in T 3R , and R ⊂ Ω is a maximal dyadic subrectangle of Ω in the form of R I × J, where I and J are cubes in 2 so that the sum converges in the T 1 2 norm.Moreover, if one assumes that f ∈ T 2 2 , then the sum also converges in the T 2  2 norm.
Proof.See 5, Proposition 3.3 for the proof.

Some Results on Product Spaces
We recall that the strong maximal function is defined as follows: where I and J are cubes in R n .It is well known that the operator M s is bounded on L p R 2n , for 1 < p < ∞.Now for any open set Ω ⊂ R n × R n with finite measure, we set By the strong maximal theorem, | Ω| ≤ c|Ω|.Denote by m 1 Ω the dyadic subrectangles R I × J ⊂ Ω that are maximal in the x 1 direction, where I and J are dyadic cubes in R n .Define m 2 Ω similarly.It is well known that Journé's covering lemma holds see 8, 15 .
where c δ is a constant depending only on δ, but not on Ω.
The following lemma shows that in order to prove that an operator is bounded from , we just need to check that the operator is uniformly bounded on the L-atoms.Lemma 2.4.Assume that T is either a linear operator or a positive sublinear operator, bounded on L 2 R 2n and for every L-atom a, with constant c independent on a. Then T can extend to a bounded operator from Proof.Its proof is similar to that of 16, Lemma 3.3 and we omit it here.See also 17 .

Some Properties of the Schr ödinger Operator L on R n
Let L be the Schr ödinger operator as 1.1 , and let h t x, y be the kernels of the operators of semigroup {e −tL }.
First we note that, for each k ∈ N, there exist two positive constants C k and c k such that the time derivatives of h t satisfy for all t > 0 and almost all x, y ∈ R n .For the proof, see, for example, 18, 19 .Next, for s > 0, we define Then for any nonzero function ψ ∈ F s , we have that where κ { ∞ 0 |ψ t | 2 dt/t} 1/2 .As an application, we have S L f 2 c f 2 for some constant c.Lemma 2.5.Let f ∈ L 2 R n and u e −t √ L f. Then for any p > 0, there exists a constant C C n, p > 0 such that where B x 0 , t 0 , r B r x 0 × t 0 − cr, t 0 with t 0 > 2cr and c > 0.
Proof.For the proof, we refer to 7, Lemma 8.4 .
Recall that if L is a nonnegative, self-adjoint operator on L 2 R n , and E L λ denotes its spectral decomposition, then for every bounded Borel function F : 0, ∞ → C, one defines the operator F L : In particular, the operator cos t √ L is then well defined on L 2 R n .Moreover, it follows from 21, Theorem 3 that the Schwartz kernel

2.13
See also 22 .By the Fourier inversion formula, whenever F is an even bounded Borel function with the Fourier transform of F, F ∈ L 1 R , we can write F √ L in terms of cos t √ L .In fact, using 2.12 we have which, when combined with 2.13 , gives , and set ϕ x 2φ 2x − φ x and ϕ t x 1/t ϕ x/t for t > 0. Let Φ and Φ t denote the Fourier transform of ϕ and ϕ t ,respectively.Then, the kernels of the operators Proof.For the proof, we refer the reader to 7, Lemma 3.5 .

Lemma 2.7. Let Ψ x
x 2 Φ x as in Lemma 2.6; then the operator

Journal of Function Spaces and Applications
Proof.For the proof, we refer to 5, Lemma 3.4 .

The Inclusion of H
We start with a suitable version of the Calder ón reproducing formula.Let Φ be as in Lemma 2.6, Ψ x : x 2 Φ x , and let c Ψ be a constant such that c Ψ ∞ 0 tΨ t e −t 2 dt 1.By L 2 -functional calculus 23 , one can write where the integral converges in L 2 R 2n .By Proposition 2.2, F has a T 1 2 -atomic decomposition: , and A k are T 1 2 -atoms associated to an open set Ω k .It is easy to see that the sum converges in T 1 2 and T 2 2 .We have Here c 0 is a large constant determined later.Using Lemma 2.7, we can show that the sum 3.

Journal of Function Spaces and Applications 9
To continue, we write

3.4
if we choose c 0 large enough.By a similar argument, we have

The Inclusion of H
By Lemma 2.4, it is enough to show that S L a 1 ≤ C for any L-atom a associated to an open set Ω.For R I × J ∈ m Ω , we let l be the maximal dyadic cube such that I ⊆ l and l × J ⊆ Ω.
Also we let S be the maximal dyadic cube such that J ⊆ S and l × S ⊆ Ω.Let R 100l × 100S.We can see that Due to H ölders inequality one has

Let us prove
R∈m Ω R c S L a x dx ≤ C. One can write

3.8
We only estimate the term I since the proof of the term II is similar.Observe that 3.9 Consider term I 1 .By Hölder's inequality, we obtain

3.10
Consider the term I 11 .It follows from estimate 2.8 that

3.11
Let x I denote the center of cube I.Note that x 1 / ∈ 100l and |x 1 −y 1 | < t 1 < l I .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 L-atom to obtain

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.
Here we omit the details.
Step II.
Step III.
By L 2 -functional calculus 23 , we have

4.1
Therefore, Step IV.H 1 max,P R n × R n ⊆ H 1 N P R n × R n , for any y 1 , y 2 , t 1 , t 2 ∈ Γ x , and x ∈ R n × R n .Applying Lemma 2.5 with 0 < p < 1, we obtain

4.3
Therefore, This completes the proof of Lemma 4.1.Let f and g be the functions of L 2 R n , and suppose that φ ∈ C ∞ 0 R n is radial and supp φ ⊆ B 0, 1 .Let u x, t e −t √ L f x .Then one has where Ψ ∈ C ∞ 0 R n with Ψ 0 is a vector-value function independent on f and g and supp Ψ ⊆ B 0, 1 .
Proof.We note that

4.6
Following the steps in 24 , we obtain

4.7
Then the lemma follows readily.
Applying Lemma 4.1, we can obtain the following lemma.
Lemma 4.2.Suppose that f, g ∈ L 2 R n × R n .Let φ and Ψ be functions as in Lemma 4.1.Then one has 4.8 Proof.The proof of Lemma 4.2 can be obtained by iterating Lemma 4.1.
We begin to show H 1 4.9 Once the claim holds, we integrate α from 0 to ∞ to complete the proof of H

3 . 13 which
, together with estimate of I 11 , show that I 1 ≤ C. By a similar argument as mentioned previously, we can show I 2 ≤ C. We have obtained the required estimate S L a 1 ≤ C.This completes the proof of Theorem 1.1.
1 N P R n × R n ⊆ H 1 S P R n × R n .Now we turn to prove the claim.The boundedness of the strong maximal operator M s on L 2 R 2n implies x | M s χ {u * >α} x ≥ Consider the term II.If g x 1 , • * Ψ t 2 y 2 / 0, then there exists x 2 , such that u * x 1 , x 2 ≤ α and |x 2 − y 2 | < t 2 .Thus, e −t 2 √L f x 1 , • y 2 ≤ α.We obtain II ≤ Cα 2 R n R n 1 g x 1 , • * Ψ t 2 y 2 2 dy 2 dt 2 t 2 dx 1 Cα 2 |{u * > α}|.For the term IV, it follows from the fact f x ≤ u * x thatStep V.H 1 S P R n × R n ⊆ H 1 atom R n × R n .It is similar to the proof of H 1 L R n × R n ⊂ H 1 atom R n × R n .We omit the details.