Commutators of certain fractional type operators with H\"ormander conditions, one-weighted and two-weighted inequalities

TL;DR

This paper investigates commutators of fractional integral operators with kernels satisfying fractional size and Hörmander conditions, establishing weighted inequalities and endpoint estimates for these operators.

Contribution

It introduces new weighted Coifman estimates and two-weight inequalities for a class of fractional type operators with complex kernels.

Findings

Established weighted $L^p(w^p)$ - $L^q(w^q)$ estimates.

Proved weighted BMO estimates for commutators.

Derived two-weight strong and endpoint estimates for the operators.

Abstract

In this paper we study the commutators of fractional type integral operators. This operators are given by kernels of theform $$K(x,y)=k_1(x-A_1y)k_2(x-A_2y)\dots k_m(x-A_my),$$ where $A_i$ are invertibles matrices and each $k_i$ satisfies a fractional size condition and generalized fractional H\"ormander condition. We obtain weighted Coifman estimates, weighted $L^p(w^p)$ - $L^q(w^q)$ estimates and weighted BMO estimates. We also give a two weight strong estimate for pair of weights of the form $(u,Su)$ where $u$ is an arbitrary non-negative function and $S$ is a maximal operator depending on the smoothness of the kernel $K$. For the singular case we also give a two-weighted endpoint estimate.