Necessary and sufficient conditions for boundedness of commutators of the general fractional integral operators on weighted Morrey spaces

TL;DR

This paper establishes a precise characterization of the boundedness of commutators of fractional integral operators on weighted Morrey spaces, linking it to Lipschitz continuity of the function involved.

Contribution

It provides necessary and sufficient conditions for the boundedness of commutators of general fractional integrals on weighted Morrey spaces, connecting operator boundedness with Lipschitz space membership.

Findings

Boundedness characterized by Lipschitz space membership

Conditions involve weighted Morrey space parameters and weight class

Results extend understanding of fractional integral commutators

Abstract

We prove that $b$ is in $Lip_{\bz}(\bz)$ if and only if the commutator $[b,L^{-\alpha/2}]$ of the multiplication operator by $b$ and the general fractional integral operator $L^{-\alpha/2}$ is bounded from the weighed Morrey space $L^{p,k}(\omega)$ to $L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega)$, where $0<\beta<1$, $0<\alpha+\beta<n, 1<p<{n}/({\alpha+\beta})$, ${1}/{q}={1}/{p}-{(\alpha+\beta)}/{n},$ $0\leq k<{p}/{q},$ $\omega^{{q}/{p}}\in A_1$ and $ r_\omega> \frac{1-k}{p/q-k},$ and here $r_\omega$ denotes the critical index of $\omega$ for the reverse H\"{o}lder condition.