Journal of Function Spaces
Volume 2015 (2015), Article ID 823862, 11 pages
http://dx.doi.org/10.1155/2015/823862

Research Article

The Boundedness of Some Integral Operators on Weighted Hardy Spaces Associated with Schrödinger Operators

Hua Wang

College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

Received 21 July 2015; Accepted 22 October 2015

Academic Editor: Dashan Fan

Copyright © 2015 Hua Wang. 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.

Let be a Schrödinger operator acting on , , where is a nonnegative locally integrable function on . In this paper, we will first define molecules for weighted Hardy spaces associated with and establish their molecular characterizations. Then, by using the atomic decomposition and molecular characterization of , we will show that the imaginary power is bounded on for , and the fractional integral operator is bounded from to , where , , and .

1. Introduction

Let and be a nonnegative locally integrable function defined on , not identically zero. We define the form by

with domain , where

It is well known that this symmetric form is closed. Note also that it was shown by Simon [1] that this form coincides with the minimal closure of the form given by the same expression but defined on (the space of functions with compact supports). In other words, is a core of the form .

Let us denote by the self-adjoint operator associated with . The domain of is given by

Formally, we write as a Schrödinger operator with potential . Let be the semigroup of linear operators generated by and let be their kernels. Since is a locally integrable nonnegative function on , then the Feynman-Kac formula implies that the semigroup kernels associated with satisfy the estimates:

for all and all .

Since the Schrödinger operator is a self-adjoint and positive definite operator acting on , then admits the following spectral decomposition:

where denotes its spectral resolution. For any , we can define the imaginary power associated with by the formula

By the functional calculus for , we can also define the operator as follows:

By the spectral theory, we know that for all . Moreover, it was proved by Shen [2] that is a Calderón-Zygmund operator provided that (reverse Hölder class). We refer the reader to [2–4] for related results concerning the imaginary powers of self-adjoint operators.

For any , the fractional integrals associated with are defined by

Since the kernel of satisfies the Gaussian upper bounds (4), then it is easy to check that for all , where denotes the classical fractional integral operator (see [5]):

Hence, by using the - boundedness of (see [5]), we have

where and . For more information about the fractional integrals associated with some classes of operators, one can see [6–9].

In [10], Song and Yan introduced the weighted Hardy space associated with in terms of the square function and established its atomic decomposition theory. Furthermore, they also showed that the Riesz transform associated with is bounded on for and bounded from to the classical weighted Hardy space (see [11, 12] for ).

Recently, in [13], we defined the weighted Hardy spaces associated with for and gave their atomic decompositions. We also obtained that is bounded from to the classical weighted Hardy space (see also [11, 12] for ) when . In this paper, we first define molecules for the weighted Hardy spaces associated with and then establish their molecular characterizations. As applications of the molecular characterization combining with the atomic decomposition of , we will obtain some estimates of and on for . Our main results are stated as follows.

Theorem 1. Let , , and . Then, for any , the imaginary power is bounded from to the weighted Lebesgue space .

Theorem 2. Let , , and . Then, for any , the imaginary power is bounded on .

Theorem 3. Suppose that . Let , , , and . Then, the fractional integral operator is bounded from to .

Theorem 4. Suppose that . Let , , , and . Then, the fractional integral operator is bounded from to .

2. Notations and Preliminaries

Let us first recall some standard definitions and notations. The classical weight theory was first introduced by Muckenhoupt in the study of weighted boundedness of Hardy-Littlewood maximal functions in [14]. A weight is a nonnegative, locally integrable function defined on , and denotes the ball with the center and radius . We say that , if

where is a positive constant which is independent of . A weight function is said to belong to the reverse Hölder class , if there exist two constants and such that the following reverse Hölder inequality holds:

Given a ball and , denotes the ball with the same center as whose radius is times that of . For a given weight function and a measurable set , we denote the Lebesgue measure of by and the weighted measure of by , where .

We give the following results which will be often used in the sequel.

Lemma 5 (see [15]). Let . Then, for any ball , there exists an absolute constant such that

In general, for any , we have

where does not depend on nor on .

Lemma 6 (see [15]). Let . Then, there exists a constant such that

for any measurable subset of a ball .

Given a weight function on , for , we denote by the weighted space of all functions satisfying

In particular, when equals a constant function, we will denote simply by and define

Throughout this paper, we will use to denote a positive constant, which is independent of the main parameters and not necessarily the same at each occurrence. By , we mean that there exists a constant such that . Moreover, we denote the conjugate exponent of by

3. Atomic Decomposition and Molecular Characterization of Weighted Hardy Spaces

Let . For any , we define and

We denote simply by when . First note that Gaussian upper bounds carry over from heat kernels to their time derivatives.

Lemma 7 (see [16, 17]). For every , there exist two positive constants and such that the kernel of the operator satisfies

for all and almost all .

Let . For any , we set

For a given function , we consider the square function associated with Schrödinger operator , which is defined by (see [18, 19])

Set

where stands for the range of in . Given a weight function on , in [10, 13], the authors defined the weighted Hardy spaces for as the completion of in the norm given by the -quasinorm of square functions; that is,

For the Schrödinger operator , it can be shown that (see, e.g., [19]). In [10], Song and Yan characterized weighted Hardy spaces in terms of atoms in the following way and obtained their atomic characterizations.

Definition 8 (see [10]). Let and . A function is called a -atom with respect to (or a --atom) if there exist a ball and a function such that

(a);(b), ;(c), .

Theorem 9 (see [10]). Let and . If , then there exist a family of --atoms and a sequence of real numbers with such that can be represented in the form , and the sum converges in the sense of both -norm and -norm.

Similarly, in [13], we introduced the notion of weighted atoms for () and proved their atomic characterizations.

Definition 10 (see [13]). Let and . A function is called a -atom with respect to (or a --atom) if there exist a ball and a function such that

;, ;, .

Theorem 11 (see [13]). Let and .

(i)If and , then there exist a family of --atoms and a sequence of real numbers with such that can be represented in the form , and the sum converges in the sense of both -norm and -norm.(ii)If and , then every --atom is in and its -norm is uniformly bounded; that is, there exists a constant independent of such that .

For every bounded Borel function , we define the operator by the following formula:

where is the spectral decomposition of . Therefore, the operator is well defined on . Moreover, it follows from [20] that there exists a constant such that the Schwartz kernel of has support contained in . By the functional calculus for and Fourier inversion formula, whenever is an even bounded Borel function with , we can write in terms of . More precisely

which gives

Lemma 12 (see [10, 19]). Let be even and . Let denote the Fourier transform of . Then, for each and for all , the Schwartz kernel of satisfies

For a given real number , we define

Then, for any nonzero function , we can prove the following estimate (see [10, 19]):

where . Inspired by [19, 21, 22], we are now going to define the weighted molecules corresponding to the weighted atoms mentioned above.

Definition 13. Let , , and . A function is called a --molecule associated with , if there exist a ball and a function such that

(A);(B), ;(C), , .

Clearly, for every --atom , it is also a --molecule for all . Then, we are able to establish the following molecular characterizations for the weighted Hardy spaces   () associated with .

Theorem 14. Let , , and .

(i)If and , then there exist a family of --molecules and a sequence of real numbers with such that , and the sum converges in the sense of both -norm and -norm.(ii)Assume that and . Then, every --molecule is in . Moreover, there exists a constant independent of such that .

Proof. (i) This is a straightforward consequence of Theorems 9 and 11. (ii) We will use some ideas from [19, 22]. Suppose that is a --molecule associated with a ball . Let be even with and let denote the Fourier transform of . We set , . By the -functional calculus of , for every , we can establish the following version of the Calderón reproducing formula:

where the above equality holds in the sense of -norm. Set

Then, we can decompose

For any measurable set in , we denote the characteristic function of the set . Hence, by formula (30), we are able to write

Let us first consider the terms . We will show that each is a multiple of a --atom with a sequence of coefficients in . Indeed, for every , one can write

where

By using Lemma 12, we can easily conclude that, for every , . In addition, by the duality argument, we get

Then, it follows from Hölder’s inequality and estimate (29) that

Hence,

which implies our desired result. Next we consider the terms . For every , we write

To deal with the term , we recall that for some , and then

where

Since , then we can further write

where

By using Lemma 12 again, we have and for every . Moreover, it follows from Minkowski’s integral inequality that

When , then . Since , then, by using Lemma 6, we can get

Consequently,

On the other hand,

Observe that . Thus, from the above discussions, we have already proved that each is a multiple of a --atom with a sequence of coefficients in . Finally, we estimate the terms . For every , we decompose as follows:

where

It follows immediately from Lemma 12 that for every and . Moreover,

Therefore, we have showed that each is also a multiple of a --atom with a sequence of coefficients in . This completes the proof of Theorem 14.

4. Proof of Theorems 1 and 2

Proof of Theorem 1. By the known result, we have that, for any , the operator is linear and bounded on (see, e.g., [2, 23]). Since is dense in , then, by Theorems 9 and 11 and a standard density argument, it is enough for us to show that, for any --atom , , there exists a constant independent of such that . Let be a --atom associated with a ball , . We write

We set . Note that , and then it follows from Hölder’s inequality, the boundedness of , and Lemma 5 that

On the other hand, for any , , by expression (7), we can write

For the term I, we observe that when , , then . Hence, by using Hölder’s inequality and estimate (4), we deduce

So we have

We now turn to estimate the other term II. In this case, since there exists a function such that and , then it follows from Hölder’s inequality and Lemma 7 that

Consequently,

where in the last inequality we have used the fact that . Therefore, by combining the above estimates for I and II, we obtain the following pointwise inequality:

Notice that . Substituting inequality (58) into the term and using Lemma 5, then we have

where the last series is convergent since . Summarizing estimates (52) and (59) derived above, we complete the proof of Theorem 1.

Proof of Theorem 2. Since is dense in and the operator is linear and bounded on , then, in view of Theorems 9, 11, and 14, it suffices to verify that, for every --atom , the function is a multiple of a --molecule for some , and the multiplicative constant is independent of . Let be a --atom with . By the definition, there exists a function such that

We set , and then . Obviously, we have . Moreover, for , we can deduce

It remains to estimate for , . We write

As mentioned in the proof of Theorem 1, we know that when , , then , . It follows from Hölder’s inequality and estimate (4) that

Since and , , then, by using Lemma 6, we can get

Hence,

Applying Hölder’s inequality and Lemma 7, we obtain

It follows immediately from inequality (64) that

Combining the above estimates for and , we thus obtain

Observe that . If we set , then we have . Therefore, we have proved that the function is a multiple of a --molecule. This completes the proof of Theorem 2.

5. Proof of Theorems 3 and 4

Proof of Theorem 3. As in the proof of Theorem 1, assuming first that , we are going to prove that, for every --atom associated with a ball , , there exists a constant independent of such that . We write

By our assumption , , then we are able to choose a number such that . Set . By a simple calculation, we can see that

Applying Hölder’s inequality with exponent , the - boundedness of (see (10)), Lemma 5, and , we get

We now turn to deal with . For any , , by expression (8), we can write

For the term III, it follows immediately from (54) that

For the other term IV, by the previous estimate (56), we thus have

where the last inequality holds since . Combining the above estimates for III and IV, we obtain the following pointwise inequality:

Note that . Substituting inequality (75) into the term , we get

Using Hölder’s inequality with exponent , Lemma 5, and the condition , we can deduce that

where in the last inequality we have used the fact that . By combining inequality (77) with (71), we obtain the desired result. Therefore, by a standard density argument, we can show that the same conclusion holds for all . This concludes the proof of Theorem 3.

Proof of Theorem 4. As in the proof of Theorem 2, it is enough to show that, for every --atom , function is a multiple of a --molecule for some and the multiplicative constant is independent of . Let be a --atom with and , where , . Setting , then we have . It is easy to check that . As before, since , then we may choose a number such that . For , by using Hölder’s inequality and the - boundedness of , we obtain

Noting that , then a straightforward computation yields that whenever . By our assumption , then we have . Consequently,

which implies

Substituting inequality (80) into (78), we can get

It remains to estimate for , . We write

For the term , by using the same arguments as in the proof of (63), we have

For the term , we proceed as that of (66) and then obtain

Combining the above estimates for and , we can get

Since , then it follows from inequality (64) that

Similar to the proof of (80), we can also show that

Hence, we finally obtain

Therefore,

Observe that . If we set , then . Summarizing estimates (81) and (89) derived above, we conclude the proof of Theorem 4.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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