Some estimates for commutators of fractional integrals associated to operators with Gaussian kernel bounds on weighted Morrey spaces

TL;DR

This paper investigates the boundedness of commutators of fractional integrals associated with operators having Gaussian kernel bounds on weighted Morrey spaces, extending understanding of their behavior in harmonic analysis.

Contribution

It provides new boundedness results for commutators of fractional integrals linked to Gaussian kernel operators on weighted Morrey spaces, with symbols in BMO or Lipschitz spaces.

Findings

Boundedness of commutators on weighted Morrey spaces established.

Results extend previous work to operators with Gaussian kernel bounds.

Analysis includes symbols in BMO and Lipschitz spaces.

Abstract

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(\mathbb R^n)$ with Gaussian kernel bound, and let $L^{-\alpha/2}$ be the fractional integrals of $L$ for $0<\alpha<n$. In this paper, we will obtain some boundedness properties of commutators $\big[b,L^{-\alpha/2}\big]$ on the weighted Morrey spaces $L^{p,\kappa}(w)$ when the symbol $b$ belongs to $BMO(\mathbb R^n)$ or homogeneous Lipschitz space.