Research Article | Open Access
Weighted Weak Local Hardy Spaces Associated with Schrödinger Operators
We characterize the weighted weak local Hardy spaces related to the critical radius function and weights which locally behave as Muckenhoupt’s weights and actually include them, by the atomic decomposition. As an application, we show that localized Riesz transforms are bounded on the weighted weak local Hardy spaces.
The theory of classical local Hardy spaces, originally introduced by Goldberg , plays an important role in various fields of analysis and partial differential equations; see [2–7] and their references. Huy Qui  studied the weighted version of the local Hardy spaces considered by Goldberg, where the weighted belongs to the Muckenhoupt class. In , Rychkov introduced and studied some properties of the weighted Besov-Lipschitz spaces and Triebel-Lizorkin spaces with weights that are locally in (Muckenhoupt’s weights, see [4, 9–11]) but may grow or decrease exponentially. In , Tang established the weighted atomic decomposition characterization of the weighted local Hardy space with local weights. Recently, in , the authors established weighted atomic decomposition characterizations for weighted local Hardy spaces with .
On the other hand, the weak space theory was first introduced by Fefferman and Soria in . Then the weak () space theory was studied by Liu in . Recently, Tang  established the weighted weak local Hardy space with local weights.
The purpose of this paper is twofold. The first goal is to characterize weighted weak local Hardy spaces by atomic decomposition. The second goal is to show that localized Riesz transforms are bounded on weighted weak local Hardy spaces.
The paper is organized as follows. In Section 2, we introduce some notation and properties concerning weights and grand maximal functions. In Section 3, we establish weighted atomic decomposition of weighted weak local Hardy spaces with . Finally, in Section 4, we show that localized Riesz transforms are bounded on weighted weak local 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 no more than . We also set and . The multiindex notation is usual: for and .
In this section, we review some notions and notations concerning the weight classes introduced in [17–19]. 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 . Particulalry, we will denote by and by .
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 that for every ball . Trivially, provided that . It is well known that if for some , then there exists , which depends only on and the constant in above inequality such that (see ). Moreover, the measure satisfies the doubling condition:
With regard to the Schrödinger operator , we know that the operators derived from behave “locally” quite similar to those corresponding to the Laplacian (see [21, 22]). The notion of locality is given by the critical radius function. Consider Throughout the paper we assume that , so that (see ). In particular, with and with .
Lemma 1 (see ). There exist and so that for all In particular, when and , where is a positive constant.
A ball of the form is called critical, and in what follows we will call any positive continuous function that satisfies (3) critical radius function, not necessarily coming from a potential . Clearly, if is such a function, so it is for any . As a consequence of the above lemma we acquire the following result.
Lemma 2 (see ). There exists a sequence of points , , such that the family , , satisfies the fact that (1);(2)for every there exist constants and such that .
In this paper, we write , where ; and denote the center and radius of , respectively.
A weight always refers to a positive function which is locally integrable. As in , we say that a weight belongs to the class for if there is a constant such that for all balls . One has We also say that a nonnegative function satisfies the condition if there exists a constant such that where When , we denote by (the standard Hardy-Littlewood maximal function). It is easy to see that for and any .
Clearly, the classes are increasing with , and we denote . By Hölder’s inequality, we see that , if , and we also denote . In addition, for , denote by the adjoint number of ; that is, .
Since with , then for , where denotes the classical Muckenhoupt weights; see [10, 24]. Moreover, the inclusions are proper. In fact, as the example given in , let and ; it is easy to check that and is not a doubling measure, but provided that and .
In what follows, given a Lebesgue measurable set and a weight , will denote the Lebesgue measure of and . For any , the space with denotes the set of all measurable functions such that and . The symbol denotes the set of all measurable functions such that We define the local Hardy-Littlewood maximal operator by
We remark that balls can be replaced by cubes in definition of and , since . In fact, for the cube , we can also define .
Next, we give some properties of weights class for .
Lemma 3. Let for . Then (i)if , then ;(ii) if and only if , where ;(iii)if ; , then there exists such that ;(iv)let , , then ;(v)let , then if and only if , where ;(vi)for , and , there exists a positive constant such that (vii)if and , then the local Hardy-Littlewood maximal operator is bounded on ;(viii)if , then is bounded from to .
For any , define the critical index of by Obviously, . If , then , but for any .
The symbols , are the dual space of . For any , let for and for .
We now introduce some local maximal functions. For and , let
Definition 5. Let and . For any , the local nontangential grand maximal function of is defined by setting, for all , and the local vertical grand maximal function of is defined by setting, for all ,
For convenience’s sake, when , we denote , , and simply by , , and , respectively; when (in which , , and are defined as in Lemmas , , and in ), we denote , , and simply by , , and , respectively. For any and , obviously,
Definition 6. Let For every , there exists an integer satisfying , and then for , and , let .
The local vertical maximal function of associated with is defined by setting, for all , the local tangential Peetre-type maximal function of associated with is defined by setting, for all , Obviously, for any , we have .
For , , and , let and for the operator , we have the following lemma.
Lemma 7 (see ). Let and ; then there exist constants and such that, for all , for all .
Proposition 8. Let . Then (i)there exists a positive constant such that, for all and almost every ,(ii)if with , , and , then
Proof. The proof of (i) is trivial. For (ii), since , it is easy to see that Hence, it suffices to prove that there exists a positive constant such that By (3.19) of , for any , satisfying and , we have By (3.28) of , we know that Therefore, by (26), (27), and , to get (25), it suffices to prove that for and there exists a constant depending only on such that We first prove (28). For any , we set , and if , otherwise is zero. Without loss of generality, we can assume that . By the boundedness of (see (vii) of Lemma 3) and the fact that , we get Next we prove (29). By Lemma 7, we have when is taken to be sufficiently large. Then by (31) and the same method of proof of (28), we obtain (29). The proof of lemma is complete.
3. The Decomposition Theorem
Let , , , and with as in (12). The weighted local Hardy spaces can be defined by and . For , we have following lemma.
Lemma 9 (see ). Let , then what follows are equivalent: (i);(ii) and ;(iii) and ;(iv) and . Moreover, for all , where the implicit constants are independent of .
Similarly, the weighted weak local Hardy spaces can be defined by and .
In this section, we establish a decomposition theorem of weighted weak local Hardy spaces .
Let and be as in (12). Throughout this section, we consider a distribution so that for all For a given , we set It is obvious that is a proper open subset of . As in , we give the usual Whitney decomposition of . Thus we can find closed cubes with , and their interiors are away from and In what follows, fix and , and if we denote , , we have . Moreover, , and have the bounded interior property; namely, every point in is contained in at most a fixed number of .
Now we take a function such that , , and on . For , set , where, and in what follows, is the center of the cube and is its side length. Obviously, by the construction of and , for any , we have , where is a fixed positive integer independent of . Let Then form a smooth partition of unity for subordinate to the locally finite covering of ; namely, with each supported in .
Let be some fixed integer and denote the linear space of polynomials in variables of degrees no more than . For each and , setThen it is easy to see that is a finite dimensional Hilbert space. Let , since induces a linear functional on via by the Riesz representation theorem; there exists a unique polynomial for each such that, for all , For each , define the distribution when (where and is the center of the cube ) and when .
As in , we can show that for suitable choices of and , the series converge in , and, in this case, we define in . We point out that the representation , where and are as above, is called a Calderón-Zygmund decomposition of of degree and height associated with .
Lemma 10. There exists a constant such that
Lemma 11. Suppose . Then there exist positive constants so that for
Lemma 12. Let and be as in (12). If , , and then there exists a positive constant such that, for all , , and , Moreover the series converges in and
Lemma 13. Let and be as in (12), , and . (i)If and , then and there exists a positive constant , independent of and , such that (ii)If and , then and there exists a positive constant , independent of and , such that .
Theorem 14. Let , , and , then, for any , there exists a sequence of bounded function with the following properties, in which such that , and if , one writes . (i) is in the sense of distribution.(ii)Each can be further decomposed as , where satisfy the fact that(1)each is supported in a cube with , , and , where , is a constant depending on , and denote the characteristic functions;(2) and for with , when and .
Conversely, if a distribution satisfies (i) and (ii), then .
Moreover, one has .
Proof. We first suppose , and set As above, let be the Whitney decomposition, and we write , , , and , respectively, as , , , and ; that is, if , if , and is a smooth function supported in . Then by Lemmas 10 and 11, there exists a constant such that for any where .
Then by Lemma 9 and using the similar method of proof of Lemma 12, we have Hence, converges in the sense of distributions, and we have the Calderón-Zygmund decomposition . By using the similar method of proof of Lemma 13(ii), we have . Let ; then for all and where , , and all the series converges in .
We set . By the similar method of Lemma in , if , it is easy to see that satisfy all conditions in (ii); if , we can decompose into a finite number of disjoint cubes , and the side length of each cube is between and , and then satisfy all conditions in (b). Obviously, is also in the sense of distribution.
For the converse, take and such that . Without loss of generality, we assume that and write Then we have , and we only need to prove Taking , where and are constants given in Lemma 1, then let , where and set By the properties of , we have then by (1) of (ii), we get Hence, to prove (50), it suffices to prove Then, we just need to estimate In fact, let ; we have the following estimate: where . From this, and note that , for any , we obtain when is small enough such that .
Therefore, we get which infers (54), and the proof is complete.
In this section, we will show the boundedness of localized Riesz transforms on spaces. As in [13, 26], for all , and , define localized Riesz transforms as where and, in what follows, , supported in , and if .
Lemma 15. Let be localized Riesz transforms, where ; then (i), for