Bump conditions and two-weight inequalities for commutators of fractional integrals

TL;DR

This paper introduces new two-weight bump conditions for sparse operators related to iterated commutators of fractional integrals, establishing bounds, necessity, and converse estimates under these conditions.

Contribution

It provides novel two-weight bump conditions for fractional integral commutators and explores their necessity and converse estimates.

Findings

Established new two-weight bump conditions for sparse operators

Derived bounds for iterated commutators under these conditions

Proved necessity and converse estimates for the bump conditions

Abstract

This paper gives new two-weight bump conditions for the sparse operators related to iterated commutators of fractional integrals. As applications, the two-weight bounds for iterated commutators of fractional integrals under more general bump conditions are obtained. Meanwhile, the necessity of two-weight bump conditions as well as the converse of Bloom type estimates for iterated commutators of fractional integrals are also given.