On commutators of certain fractional type integrals with Lipschitz functions

Abstract

Abstract In this paper, we study the commutators generated by Lipschitz functions and fractional type integral operators with kernels of the form Kα(x,y)=κ1(x−A1y)κ2(x−A2y)⋯κm(x−Amy), $$ K_{\alpha }(x,y) = \kappa _{1}(x - A_{1}y) \kappa _{2}(x - A_{2}y)\cdots \kappa _{m}(x - A_{m}y), $$ where 0≤α=α1+⋯+αm<n $0\le \alpha =\alpha _{1}+\cdots +\alpha _{m}< n$, each κi $\kappa _{i}$ satisfies the (n−αi) $(n-\alpha _{i})$-order fractional size condition and a generalized fractional Hörmander condition, Ai $A_{i}$ is invertible, and Ai−Aj $A_{i}-A_{j}$ is invertible for i≠j $i \neq j$, 1≤i,j≤m $1 \leq i, j \leq m$. We establish the corresponding sharp maximal function estimates and obtain the weighted Coifman type inequalities, weighted Lp(wp)→Lq(wq) $L^{p}(w^{p}) \rightarrow L^{q}(w^{q})$ estimates, and the weighted endpoint estimates for such commutators.

Keywords

• Commutators
• Lipschitz functions
• Generalized fractional Hörmander condition
• Fractional size condition
• Weighted estimates
• Coifman type inequality