Research Article | Open Access
Riesz Transform Characterization of Weighted Hardy Spaces Associated with Schrödinger Operators
We characterize the weighted local Hardy spaces related to the critical radius function and weights by localized Riesz transforms ; in addition, we give a characterization of weighted Hardy spaces via Riesz transforms associated with Schrödinger operator , where is a Schrödinger operator on () and is a nonnegative function satisfying the reverse Hölder inequality.
Let be a Schrödinger operator on , , where is a fixed nonnegative potential. We assume that belongs to the reverse Hölder class for some ; that is, there exists such thatfor every ball . With regard to the Schrödinger operator , we know that the operators derived from behave “locally” quite similar to those corresponding to the Laplacian (see [1, 2]). The notion of locality is given by the critical radius functionThroughout the paper we assume that , so that (see ).
The study of Schrödinger operator recently attracted much attention; see [2–14]. In particularly, Dziubański and Zienkiewicz [5, 6] studied Hardy space associated with Schrödinger operators . In , we introduce the weighted Hardy space with (for Muckenhoupt’s weights, see [15–19]), and in , we obtain some weighted estimates for bilinear operators on .
Recently, Bongioanni et al.  introduced new classes of weights, related to Schrödinger operators , that is, weight, which are in general larger than Muckenhoupt’s (see Section 2 for notions of weight). In [9, 10, 20], Tang gave some results about weighted norm inequalities with weights . In , we have obtained an atomic characterization for weighted Hardy space with , by introducing the weighted local Hardy spaces with weights and establishing the atomic characterization of the weighted local Hardy spaces with weights.
An interesting question is whether we can give a characterization of weighted Hardy spaces with via Riesz transforms associated with Schrödinger operator . Since it is difficult to give the characterization by the same method in [5, 8], in this paper, we will establish the characterization by a new method, which will take advantage of weighted local Hardy spaces and localized Riesz transforms. In fact, by this method, we can give the Riesz transforms characterization of at the same time.
The paper is organized as follows. In Section 2, we review some notions and notations concerning the weight classes introduced in [3, 9, 10], and we also review some results about and . In Section 3, we establish the Riesz transforms characterization of weighted Hardy spaces .
Throughout this paper, we let denote constants that are independent of the main parameters involved but whose value may differ from line to line. By , we mean that there exists a constant such that . The symbol means that . The symbol for denotes the maximal integer not more than . We also set and .
In this section, we first review some notions and notations concerning the weight classes introduced in [3, 9, 10]. Given and , we will write for the -dilate ball, which is the ball with the same center and with radius . Similarly, denotes the cube centered at with side length (here and below only cubes with sides parallel to the axes are considered), and . Particularly, we will denote by and by . The following lemma is a basic property of the critical radius function .
Lemma 1 (see ). There exist and so that for all In particular, when and , where is a positive constant.
Lemma 2 (see ). There is a positive constant and a collection of balls , , , such that , , and for every and here and in what follows, for any set , denotes its cardinality.
Lemma 3 (see ). There are nonnegative functions such that (i)for all and , ;(ii) for all ;(iii)there exists a positive constant s.t. for all , , and
In this paper, we write , where and and denote the center and radius of , respectively. As in , we say that a weight belongs to the class for , if there is a constant such that for all balls We also say that a nonnegative function satisfies the condition if there exists a constant such that , a.e. , where When , we denote by (the standard Hardy-Littlewood maximal function). It is easy to see that for a.e. and any . Clearly, the classes are increasing with , and we denote and . In addition, for , denote by the adjoint number of ; that is, .
We remark that balls can be replaced by cubes in definition of and , since In fact, for the cube , we can also define . Now, we give some properties of weights class for .
For any , define the critical index of by
For any , let for and for . Now let us introduce the space . Let be the semigroup of linear operators generated by and let be their kernels; that is, , for and A weighted Hardy-type space related to with weights is naturally defined by , with . with has been studied in [8, 14], and we have the following theorem.
Theorem A (see ). If and is a nonnegative potential, then, for all , one has
Then let us recall some results about weighted local Hardy spaces . We first introduce some local maximal functions. For and , let
For any , the local nontangential grand maximal function of is defined asand the local vertical grand maximal function of is defined as
For the sake of convenience, we denote , , and as , , and , when , and as , , and , when (where , , and are defined as in Lemmas 4.2, 4.4, and 4.8 in ). For any and , obviously,
Definition 5. Let and let be as in (7), , and For each with , the weighted local Hardy space is defined by Moreover, let .
Here and in what follows, we definewhere and are, respectively, as in Definition 2.2 and Theorem 3.1 of .
Next, we introduce the weighted local atoms and weighted atomic local Hardy space.
Definition 6. Let and let be as in (7). A triplet is called admissible, if , , and with . A function on is said to be a -atom if (i) and ,(ii),(iii) for all with , when , , where , , and and are constants given in Lemma 1. Moreover, a function on is called a -single-atom with , if
Lemma 7 (see ). Let and be a -atom, which satisfies ; then there exists a constant such that
Definition 8. Let and let be as in (7), and let be admissible. The weighted atomic local Hardy space is defined as the set of all satisfying that in , where are -atoms, is a -single-atom, and . Moreover, the quasi-norm of is defined by , where the infimum is taken over all the decompositions of as above.
In virtue of the above theorem, for notational simplicity, we denote by the weighted local Hardy space when .
Definition 9. Let and let be admissible; then is defined as the vector space of all finite linear combinations of -atoms and a -single-atom, and the norm of in is defined by , where , , , are -atoms, and is a -single-atom.
Let . A quasi-Banach space with the quasi-norm is called a -quasi-Banach space if for all . For any given -quasi-Banach space with and a linear space , an operator from to is said to be -sublinear if, for any and , and .
Theorem D (see ). Let , , and be a -quasi-Banach space. Suppose and is a -sublinear operator such that Then there exists a unique bounded -sublinear operator from to which extends .
Theorem E (see ). Let and ; then with equivalent norms; that is, .
3. Riesz Transform Characterization of Weighted Hardy Spaces
The main purpose of this section is to give a characterization of weighted Hardy spaces with via Riesz transforms associated with Schrödinger operator . We begin with some useful lemmas.
Lemma 10 (see ). If , then there is a constant such that
Lemma 11 (see ). There exists a sequence of points , , such that the family , , satisfies the following:(a).(b)For every there exist constants and such that .
For , let us define the Riesz transforms associated with Schrödinger operator by . In addition, as in , for all and , we define localized Riesz transforms aswhere and in what follows , supported in , and if . For all and , we define
With these operators, we can get the following boundedness.
Lemma 12. Let be as in (14); then (i), for and ,(ii), for .
Proof. Let ; then by [15, 18], we know that is bounded on for and bounded from to . Let be the classical Riesz transforms; then for all , we have where and in what follows denotes the Hardy-Littlewood maximal operator. By the above estimate, combining with the boundedness on for and boundedness from to of , , and , we can obtain the conclusions of the lemma.
Then for , we can obtain the following lemma.
Lemma 13. Let be as in (14); then (i), for and ,(ii), for .
Proof. By Lemma 11, there exists a sequence of points , , such that . Let , with as in Lemma 1, and set . Then by Lemma 1 of , for and any , there exists such that on . Thus, by Lemma 12, we have Summing up all balls of gives (i).
For (ii), as in (i), for and any , there exists such that on . Then, by Lemma 12, we have Summing up all balls of gives (ii).
The proof is complete.
Now we can establish the Riesz transform characterization as follows.
Theorem 14. Let and ; then one has
Proof. is obvious by Theorem E. We now prove We first assume that and here we will borrow some idea from . Let be a ball as in Lemma 2, and let be a nonnegative function supported in and on , where . By properties of weights, we can set so that on , where is a constant independent of and will be given in the following proof. Taking , by Theorem A, we haveLet us denote the integral kernel of the operator by , and let be the kernel of the classical Riesz transform . Then we can define and we also have Let ; then we have For , by the same method of the proof of Lemma 10 (Lemma 3.14 of ), we get For , by Lemma 3 of , we haveSince and , we have , and there is a constant independent of such that We now estimate . For and , we have , and there exists a constant independent of such that . On the other hand, if , then there exists , which depends only on and the constant in (3), such that (see ). Thus, according to Lemma 3 of , we get For the inner integral, there exists a constant independent of such that and for all ; then by the properties of weights, we have where satisfies . Hence, we obtain