On commutators of certain fractional type integrals with Lipschitz functions

Abstract

In this paper, we study the commutators generated by Lipschitz functions and fractional type integral operators with kernels of the form \t\t\tKα(x,y)=κ1(x−A1y)κ2(x−A2y)⋯κm(x−Amy),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ K_{\\alpha }(x,y) = \\kappa _{1}(x - A_{1}y) \\kappa _{2}(x - A_{2}y)\\cdots \\kappa _{m}(x - A_{m}y), $$\\end{document} where 0le alpha =alpha _{1}+cdots +alpha _{m}< n, each kappa _{i} satisfies the (n-alpha _{i})-order fractional size condition and a generalized fractional Hörmander condition, A_{i} is invertible, and A_{i}-A_{j} is invertible for i neq j, 1 leq i, j leq m. We establish the corresponding sharp maximal function estimates and obtain the weighted Coifman type inequalities, weighted L^{p}(w^{p}) rightarrow L^{q}(w^{q}) estimates, and the weighted endpoint estimates for such commutators.