Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

and Applied Analysis 3 2. Preliminaries Firstly, we recall some lemmas of the auxiliary function ρ(x) which have been proved by Shen in [9]. Throughout this section we always assume V ∈ B s 1 for some s 1 > (n/2). Lemma 6. The measure V(x)dx satisfies the doubling condition; that is, there exists C > 0 such that


Introduction
Let  = −Δ +  be a Schrödinger operator, where Δ is the laplacian on R  and the nonnegative potential  belongs to the reverse Holder class   1 for some  1 ≥ (/2) and  ≥ 3.In this paper, we consider the Riesz transform associated with the Schrödinger operator  as follows: R = ∇(−Δ + ) −1/2 . ( Let  be a locally integrable function on R  and let  be a linear operator.For a suitable function , the commutator is defined by    ≐ [, ] = () − ().It is well known that when  is a Calderón-Zygmund operator, Coifman et al. [1] proved that [, ] is a bounded operator on   for 1 <  < ∞ if and only if  ∈ BMO(R  ).
Recently, some scholars have investigated the boundedness of the commutators generated by a BMO function  and Riesz transforms associated with the Schrödinger operator (cf.[2][3][4][5][6][7][8]).It follows from [9] that Riesz transform associated with the Schrödinger operator  is not a Calderón-Zygmund operator if the potential  ∈   ((/2) <  < ).Their results imply that the boundedness of the commutators of Riesz transform associated with the Schrödinger operator  depends on the nonnegative potential .In [5], the authors have obtained the weighted   , 1 <  < ∞, and weak  log  the estimates for the commutator R  .In this paper we are interested in the weighted Hardy space estimates for R  , which are also the weighted endpoint estimates.It is noted that our main results generalize Theorem 2.7 and Theorem 4.1 in [3] to the weighted case and the function  that we consider belongs to a larger class than the classical BMO space.
Note that a nonnegative locally   integrable function  on R  is said to belong to   (1 <  < ∞) if there exists  > 0 such that the reverse Hölder inequality holds for every ball  in R  .Obviously, But it is important that the   class has a property of "selfimprovement"; that is, if  ∈   , then  ∈  + for some  > 0.
Throughout this paper, we set () = ∫  () for any subset  ⊆ R  .Assume that the nonnegative function for all balls  in R  , where We say that  ∈  1 (R  ), if there exists a positive constant  > 0, such that where  is a Hardy-Littlewood operator.Given a weight function  ∈   (R  ) for 1 ≤  < ∞, as usual we denote by   () the space of all measurable functions satisfying When  = ∞,  ∞ () will be taken to mean  ∞ and ‖‖  ∞ () = ‖‖  ∞ .Moreover, denote by  1 weak () the space of all measurable functions satisfying sup In the rest of this paper, we always assume that  ∈  1 (R  ).
Following the above definition of atoms and the above atomic decomposition, we know that the weighted Hardy space  1  () is not the special case of Hardy spaces established by Yang and Zhou in [12].Now we are in a position to give the main results in this paper.
Theorem 4. Let  ∈  1 (R  ).Suppose  ∈   for  > (/2).Then, where  > 0. Namely, the commutator R  is bounded from This paper is organized as follows.In Section 2, we recall some basic facts to prove main results in this paper.Section 3 gives the proof of weighted estimates of Riesz transform associated with the Schrödinger operator.In Section 4, we prove Theorem 5.
Throughout this paper, the letter  stands for a constant and is not necessarily the same at each occurrence.By  1 ∼  2 , we mean that there exists a constant  > 1 such that (1/) ≤ ( 1 / 2 ) ≤ .Moreover, for the ball  = (, ), we denote the ball  by  = (, ), where  is a positive constant.

Preliminaries
Firstly, we recall some lemmas of the auxiliary function () which have been proved by Shen in [9].Throughout this section we always assume  ∈   1 for some  1 > (/2).Lemma 6.The measure () satisfies the doubling condition; that is, there exists  > 0 such that holds for all balls (, ) in R  .
Lemma 10.There exists  1 > 0 such that For the proofs of Lemma 6 to Lemma 10, readers can refer to [9].
Thirdly, we recall some important and useful properties of   weights (cf.Chapter V in [13]).
Then there exist a constant  > 0 and  > 1 depending only on  and the   constant of , such that for any ball .
At last, we review some basic facts about the BMO space BMO  (), which have been proved in [4].

Weighted Estimates of Riesz Transforms Associated with the Schrödinger Operators
In this section, we need to prove the weighted estimates of the Riesz transform associated with the Schrödinger operator, which will be used in the proof of Theorem 5.
By [13, By duality, we have the following.
In order to prove Theorem 4, we need the following lemmas.
Proof of Theorem 4. We show Theorem 4 by a method similar to the one used in the proof of Theorem 2 in [15].By the Calderón-Zygmund decomposition in the proof of Theorem 3.5 on page 413 in [16], given  ∈  1 () and  > 0, we have  =  1 +  2 , with  2 = ∑    , such that the following hold.
where we have used Lemma 10 in the last inequality above.