Journal of Inequalities and Applications (Aug 2019)
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.
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