Weighted boundedness for Toeplitz type operator related to singular integral transform with variable Calder´on-Zygmund kernel

: In the paper, some weighted maximal inequalities for the Toeplitz operator related to the singular integral transform with variable Calder´on-Zygmund kernel are proved. As an application, the boundedness of the operator on weighted Lebesgue space are obtained.


Introduction
Suppose b is a locally integrable function on R n and T is an integral operator. The principle model of commutator generated by b and T is Calderón commutator [T, M b ]( f ) = T (b f )−bT ( f )(see [5]). The boundedness of commutator characterizes some function spaces (see [2,10,21]). In the mid seventies, Coifman, Rochberg and Weiss showed that the commutator is bounded on Lebesgue space. In fact they even proved that this property characterizes BMO functions. As the development of singular integrals (see [7,21]), the commutator has been well studied. In [5,19,20], the authors proved that the commutators of BMO functions and the singular integral are bounded on Lebesgue space. In [3], the author proved a similar result where singular integral is replaced by fractional integral. In [10,18], the boundedness of the commutator of the Lipschitz function and singular integral on Triebel-Lizorkin and Lebesgue spaces are gained. In [1,9], the boundedness for the commutator by the weighted BMO and Lipschitz functions and singular integral on Lebesgue spaces are gained (also see [8]). In [2], the authors introduced certain singular integral operator with variable kernel and obtained its boundedness. In [13][14][15], the boundedness for the commutator by the BMO function and operator is obtained. In [17], the authors proved the boundedness of the multilinear oscillatory singular integral by BMO function and the operator. In [11,12,16], certain Toeplitz operator related to the strongly singular integral is studied.
Motivated by these, in the paper, certain Toeplitz operator of the weighted BMO and Lipschitz functions with the singular integral transform with variable Calderón-Zygmund kernel are studied.

Preliminaries and notations
In the paper, we will study following singular integral transforms (see [2]) Definition. Let K(x, ·) be a variable Calderón-Zygmund kernel for a.e. x ∈ R n as [2] and for a locally integrable function b on R n and the singular integral transform T with variable Calderón-Zygmund kernel as The Toeplitz operator relater to T is defined as where T k,1 are the ±I(the identity operator) or singular integral transform with variable Calderón-Zygmund kernel, and T k,2 are the linear operators for k = 1, ..., m, M b ( f ) = b f . Now, we introduce some notations. In the paper, Q will denote a cube of R n . For a weight function ω (i.e. ω is a nonnegative locally integrable function), let ω(Q) = Q ω(x)dx and ω Q = |Q| −1 Q ω(x)dx. For a locally integrable function b, the maximal sharp function of b is defined by We know that (see [7]) The A p weight is defined by (see [7]) Remark. (1). We know that (see [6]) (2). Given f ∈ Lip β (v) and v ∈ A 1 . By [5], It is known that spaces Lip β (v) coincide and the norms The following preliminary lemma needs. Lemma 1.([7, p.485]) Suppose 0 < p < q < ∞ and any positive function f . It is defined that, for where the sup is taken for all measurable sets Q with 0 < |Q| < ∞. Then Lemma 2.(see [2]) Suppose T is the singular integral transform as Definition 2. Then T is bounded on Lemma 9.(see [7]). Suppose 0 < p, η < ∞ and v ∈ ∪ 1≤r<∞ A r . Then, for any smooth function f ,
Proof of Theorem 1. It is only to prove the following inequality holds, for f ∈ C ∞ 0 (R n ) and some constant C 0 : We assume T k,1 are T (k = 1, ..., m). Fix a cube Q = Q(x 0 , d) andx ∈ Q. We write, by T 1 (g) = 0, For I 1 , we know ν −r/p ∈ A r by Lemma 4, we get then, by Lemmas 1, 2 and 5, we obtain For I 2 , by [2], we know that |x − y| n f (y)dy, Thus, by the same argument of proof in [4], for x ∈ Q, we get Theorem 1 is proved.
Proof of Theorem 2. It only to prove the following inequality holds, for f ∈ C ∞ 0 (R n ) and some constant C 0 : We assume T k,1 are T (k = 1, ..., m) and similar to Theorem 1, for a cube Q = Q(x 0 , d) andx ∈ Q, we get For I 3 , we have For I 4 , by using the same argument as in the proof of I 2 , we have, for x ∈ Q, Theorem 2 is proved. Proof of Theorem 3. It is noticed ν r /p ∈ A r +1−r /p ⊂ A p and ν(x)dx ∈ A p/r (ν(x) r /p dx), we have, by Theorem 1 and Lemma 9, Theorem 3 is proved.
Proof of Theorem 4. In Theorem 2 we choose 1 < s < p and by v 1−q ∈ A ∞ , we get, by Lemmas 8 and 9, Theorem 4 is proved.

Conclusions
Some new weighted maximal inequalities for the Toeplitz operator related to the singular integral transform with variable Calderón-Zygmund kernel are proved. As an application, the boundedness of the operator on weighted Lebesgue space are obtained.