Weighted Weak Local Hardy Spaces Associated with Schrödinger Operators

The theory of classical local Hardy spaces, originally introduced by Goldberg [1], plays an important role in various fields of analysis and partial differential equations; see [2–7] and their references. Huy Qui [2] studied the weighted version h ω of the local Hardy spaces h considered by Goldberg, where the weighted ω belongs to the Muckenhoupt class. In [8], Rychkov introduced and studied some properties of the weighted Besov-Lipschitz spaces and Triebel-Lizorkin spaces with weights that are locally in Ap(R ) (Muckenhoupt’s weights, see [4, 9–11]) but may grow or decrease exponentially. In [12], Tang established the weighted atomic decomposition characterization of the weighted local Hardy space


Introduction
The theory of classical local Hardy spaces, originally introduced by Goldberg [1], plays an important role in various fields of analysis and partial differential equations; see [2][3][4][5][6][7] and their references.Huy Qui [2] studied the weighted version ℎ   of the local Hardy spaces ℎ  considered by Goldberg, where the weighted  belongs to the Muckenhoupt class.In [8], Rychkov introduced and studied some properties of the weighted Besov-Lipschitz spaces and Triebel-Lizorkin spaces with weights that are locally in   (R  ) (Muckenhoupt's weights, see [4,[9][10][11]) but may grow or decrease exponentially.In [12], Tang established the weighted atomic decomposition characterization of the weighted local Hardy space ℎ   (R  ) with local weights.Recently, in [13], the authors established weighted atomic decomposition characterizations for weighted local Hardy spaces ℎ   () with  ∈  ,  (R  ).On the other hand, the weak  1 space theory was first introduced by Fefferman and Soria in [14].Then the weak   (0 <  < 1) space theory was studied by Liu in [15].Recently, Tang [16] established the weighted weak local Hardy space Wh   (R  ) 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  ∈  ,  (R  ).Finally, in Section 4, we show that localized Riesz transforms are bounded on weighted weak local Hardy spaces.

Preliminaries
In this section, we review some notions and notations concerning the weight classes  ,  (R  ) introduced in [17][18][19].Given  = (, ) and  > 0, 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 2 by  * and 2 by  * .

Journal of Function Spaces
Let L = −Δ +  be a Schrödinger operator on R  ,  ≥ 3, where  ̸ ≡ 0 is a fixed nonnegative potential.We assume that  belongs to the reverse Hölder class RH  (R  ) for some  ≥ /2; that is, there exists  = (, ) > 0 such that for every ball  ⊂ R  .Trivially, RH  (R  ) ⊂ RH  (R  ) provided that 1 <  ≤  < ∞.It is well known that if  ∈ RH  (R  ) for some  > 1, then there exists  > 0, which depends only on  and the constant  in above inequality such that  ∈ RH + (R  ) (see [20]).Moreover, the measure () satisfies the doubling condition: With regard to the Schrödinger operator L, we know that the operators derived from L 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  ̸ ≡ 0, so that 0 < () < ∞ (see [22]).In particular,   () = 1 with  = 1 and Lemma 1 (see [22]).There exist  0 ≥ 1 and  0 ≥ 1 so that for all ,  ∈ R 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  > 0. As a consequence of the above lemma we acquire the following result.
A weight always refers to a positive function which is locally integrable.As in [17], we say that a weight  belongs to the class  ,  (R  ) for 1 <  < ∞ if there is a constant  such that for all balls .One has We also say that a nonnegative function  satisfies the  , .In addition, for 1 ≤  ≤ ∞, denote by   the adjoint number of ; that is, 1/ + 1/  = 1.
Similarly, the weighted weak local Hardy spaces Wh   () can be defined by In this section, we establish a decomposition theorem of weighted weak local Hardy spaces Wh   ().We first recall the Calderón-Zygmund decomposition of  of degree  and height  associated with M  () as in [12,13,25].
Let  ∈ Z + be some fixed integer and P  (R  ) denote the linear space of polynomials in  variables of degrees no more than .For each  ∈ N and  ∈ P s (R  ), set Then it is easy to see that (P  (R  ), ‖⋅‖  ) is a finite dimensional Hilbert space.Let  ∈ D  (R  ), since  induces a linear functional on P  (R  ) via by the Riesz representation theorem; there exists a unique polynomial   ∈ P  (R  ) for each  such that, for all  ∈ For each , define the distribution   ≡ ( −   )  when   ∈ (0,  3 (  )) (where  3 = 2  0  0 and   is the center of the cube   ) and   ≡   when   ∈ [ 3 (  ), ∞).
As in [13], we can show that for suitable choices of  and , the series ∑    converge in D  (R  ), and, in this case, we define  ≡  − ∑    in D  (R  ).We point out that the representation  =  + ∑    , where  and   are as above, is called a Calderón-Zygmund decomposition of  of degree  and height  associated with M  ().
To obtain the main theorem, we need the following lemmas (Lemmas 10-13) about Calderón-Zygmund decomposition which have been given in Section 4 of [13].
Then by Lemma 9 and using the similar method of proof of Lemma 12, we have where =1 satisfy all conditions in (b).Obviously,  = ∑ +∞ =    is also in the sense of distribution.