Commutator Theorems for Fractional Integral Operators on Weighted Morrey Spaces

TL;DR

This paper establishes necessary and sufficient conditions for the boundedness of commutators of fractional integral operators associated with an analytic semigroup on weighted Morrey spaces, extending classical results to weighted and Morrey contexts.

Contribution

It provides new characterizations of boundedness for commutators of fractional integrals on weighted Morrey spaces, involving weighted BMO spaces, which was not previously known.

Findings

Boundedness characterized by weighted BMO conditions

Necessary and sufficient conditions derived for weighted Morrey spaces

Extension of classical commutator results to weighted Morrey setting

Abstract

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(R^n)$ with Gaussican kernel bounds, and let $L^{-\alpha/2}$ be the fractional integrals of $L$ for $0<\alpha<n.$ For any locally integrable function $b$, The commutators associated with $L^{-\alpha/2}$ are defined by $[b,L^{-\alpha/2}](f)(x)=b(x)L^{-\alpha/2}(f)(x)-L^{-\alpha/2}(bf)(x)$. When $b\in BMO(\omega)$(weighted $BMO$ space) or $b\in BMO$, the author obtain the necessary and sufficient conditions for the boundedness of $[b,L^{-\alpha/2}]$ on weighted Morrey spaces respectively.