Weighted norm inequalities for Toeplitz type operators associated to generalized Calderón–Zygmund operators

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_1$$\end{document}T1 be a generalized Calderón–Zygmund operator or \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm I$$\end{document}±I ( the identity operator), let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_2$$\end{document}T2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_3$$\end{document}T3 be the linear operators, and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_3=\pm I$$\end{document}T3=±I. Denote the Toeplitz type operator by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} T^b=T_1M^bI_\alpha T_2+T_3I_\alpha M^b T_4, \end{aligned}$$\end{document}Tb=T1MbIαT2+T3IαMbT4,where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M^bf=bf,$$\end{document}Mbf=bf, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_\alpha $$\end{document}Iα is fractional integral operator. In this paper, we establish the sharp maximal function estimates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^b$$\end{document}Tb when b belongs to weighted Lipschitz function space, and the weighted norm inequalities of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^b$$\end{document}Tb on weighted Lebesgue space are obtained.

The operator T is called a generalized Calderón-Zygmund operator provided the following three conditions are satisfied: 1. T can be extended as a continuous operator on L 2 (R n ); 2. K is smooth away from the diagonal {(x, y) : x = y} with where C > 0 is a constant independent of y and z; 3. There is a sequence of positive constant numbers {C j } such that for each j ∈ N , and where (γ , γ ′ ) is a fixed pair of positive numbers with 1/γ + 1/γ ′ = 1 and 1 < γ ′ < 2.
If we compare the generalized Calderón-Zygmund operator with the classical Calderón-Zygmund operator, whose kernel K(x, y) enjoys the conditions and where |x − y| > 2|z − y| for some δ > 0, we can find out that the classical Calderón-Zygmund operator is a generalized Calderón-Zygmund operator defined above with C j = 2 −jδ , j ∈ N , and any 1 < γ < ∞.
Let b be a locally integrable function on R n . The Toeplitz type operator associated to generalized Calderón-Zygmund operator and fractional integral operator I α is defined by where T 1 is the generalized Calderón-Zygmund operator or ±I (the identity operator), T 2 and T 4 are the linear operators, T 3 = ±I, and M b f = bf .
Note that the commutators [b, I α ](f ) = bI α (f ) − I α (bf ) are the particular operators of the Toeplitz type operators T b . The Toeplitz type operators T b are the non-trivial generalizations of these commutators.
It is well known that the commutators of fractional integral have been widely studied by many authors. Paluszyński (1995) showed that b ∈ Lip β (R n )(0 < β < 1) (homogeneous Lipschitz space) if and only if [b, I α ] is bounded from L p (R n ) to L q (R n ), where 1 < p < n/(α + β) and 1/q = 1/p − (α + β)/n. When b belongs to the weighted Lipschitz spaces Lip β (ω), Hu and Gu (2008) L q (ω 1−(1−α/n)q ) for 1/q = 1/p − (α + β)/n with 1 < p < n/(α + β). A similar result obtained when I α is replaced by the generalized fractional integral operator (Hu et al. 2013). This paper investigates the boundedness of the Toeplitz type operator associated to generalized Calderón-Zygmund operator, fractional integral operator I α and weighted Lipschitz function on weighted Lebesgue space. The main result is as follows.
The paper is organized as follows. Section introduces some notation and definitions, and recalls some preliminary results. Section establishes the sharp estimates for Toeplitz type operators. Section gives the proof of Theorem 1.
In this paper, we shall use the symbol A B to indicate that there exists a universal positive constant C, independent of all important parameters, such that A ≤ CB. A ≈ B means that A B and B A.

Some preliminaries
A weight ω is a nonnegative, locally integrable function on R n . Let B = B r (x 0 ) denote the ball with the center x 0 and radius r, and let B = B r (x 0 ) for any > 0. For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by |E| and set weighted measure ω(E) = E ω(x)dx. For any given weight function ω on R n , 0 < p < ∞, denote by L p (ω) the space of all function f satisfying Definition 2 (Muckenhoupt 1972 where the supremum is considered over all ball B ⊂ R n and, ω ∈ A 1 if

Definition 3 (Muckenhoupt and Wheeden 1974) A weight function ω belongs to
From the definition of A p,q , we can get that Definition 4 (García-Cuerva and Rubio de Francia 1985) A weight function ω belongs to the reverse Hölder class RH s if there exists constant s > 1 such that the following reverse Hölder inequality holds for every ball B ⊂ R n .
It is well known that if ω ∈ A p with 1 < p < ∞, then ω ∈ A r for all r > p, and ω ∈ A q for some 1 < q < p. If ω ∈ A p with 1 ≤ p < ∞, then there exists r > 1 such that ω ∈ RH r . It follows directly from Hölder's inequality that ω ∈ RH r implies ω ∈ RH s for all 1 < s < r. Moreover, if ω ∈ RH r , r > 1, then we have ω ∈ RH r+ǫ for some ε > 0. We write r ω = sup{r > 1 : ω ∈ RH r } to denote the critical index of ω for the reverse Hölder condition.

Lemma 1 (García-Cuerva and Rubio de Francia 1985)
The following results about weight function are right.
Then, for any ball B and any > 1, Next, we shall recall the definition of the Hardy-Littlewood maximal operator and several variants, the fractional integral operator and some function spaces.

Definition 5 The Hardy-Littlewood maximal operator Mf is defined by
.
The sharp maximal operator M ♯ f is defined by Lemma 2 (Stein 1993 Definition 6 For 0 ≤ α < n, t ≥ 1, we define the fractional maximal operator M α,t f by and define the fractional weighted maximal operator M α,r,ω f by In order to simplify the notation, we set M α = M α,1 , M t,ω = M 0,t,ω .
Definition 7 For 0 < α < n, the fractional integral operator I α is defined by Lemma 3 Let I α be fractional integral operator, and let E be a measurable set in R n .
Then for any f ∈ L 1 (R n ), we have Let us recall the weighted Lipschitz function space.
Definition 8 For 1 ≤ p < ∞, 0 < β < 1, and ω ∈ A ∞ . A locally integrable function b is said to be in the weighted Lipschitz function space if where b B = |B| −1 B b(y)dy, and the supremum is taken over all balls B ⊆ R n . The Banach space of such functions modulo constants is denoted by Lip β,p (ω). The smallest bound C satisfying conditions above is then taken to be the norm of b denoted by b Lip β,p (ω) . Obviously, for the case ω = 1, the Lip β,p (ω) space is the classical Lip β (R n ) space. Put Lip β (ω) = Lip β,1 (ω). Let ω ∈ A 1 . Garcia-Cuerva (1979) proved that the spaces Lip β,p (ω) coincide, and the norms b Lip β,p (ω) are equivalent with respect to different values of p provided that 1 ≤ p < ∞. Since we always discuss under the assumption ω ∈ A 1 in the following, then we denote the norm of Lip β,p (ω) by � · � Lip β (ω) for 1 ≤ p < ∞.

The sharp estimates for T b
To prove our main result, we first prove the following the sharp estimates for T b .

Then
Combining the estimates for M 1 , M 2 , M 3 and M 4 , the proof of Theorem 2 is completed.