Abstract
Let T_1 be a generalized Calderón–Zygmund operator or pm I ( the identity operator), let T_2 and T_3 be the linear operators, and let T_3=pm 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 M^bf=bf, and I_alpha is fractional integral operator. In this paper, we establish the sharp maximal function estimates for T^b when b belongs to weighted Lipschitz function space, and the weighted norm inequalities of T^b on weighted Lebesgue space are obtained.
Highlights
Introduction and resultsAs the development of the singular integral operators, their commutators have been well studied (Coifman et al 1976; Harboure et al 1997; Lin et al 2015)
Coifman et al (1976) proved that the commutators [b, T], which generated by Calderón–Zygmund singular integral operators and BMO functions, are bounded on Lp(Rn) for 1 < p < ∞
Chanillo (1982) obtained a similar result when Calderón–Zygmund singular integral operators are replaced by the fractional integral operators
Summary
Let us recall the weighted Lipschitz function space. A locally integrable function b is said to be in the weighted Lipschitz function space if sup ω (B)β/n. Where bB = |B|−1 B b(y)dy, and the supremum is taken over all balls B ⊆ Rn. The Banach space of such functions modulo constants is denoted by Lipβ,p(ω). The smallest bound C satisfying conditions above is taken to be the norm of b denoted by b Lipβ,p (ω). Garcia-Cuerva (1979) proved that the spaces Lipβ,p(ω) coincide, and the norms b Lipβ,p (ω) are equivalent with respect to different values of p provided that 1 ≤ p < ∞. Since we always discuss under the assumption ω ∈ A1 in the following, we denote the norm of Lipβ,p(ω) by · Lipβ(ω) for 1 ≤ p < ∞.
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