Endpoint estimates for commutators of sublinear operators in the Morrey type spaces

TL;DR

This paper establishes endpoint estimates for commutators of sublinear operators with BMO functions in Morrey-type spaces, extending the boundedness theory in harmonic analysis.

Contribution

It provides new endpoint estimates for these commutators in weighted and generalized Morrey spaces, under minimal assumptions.

Findings

Endpoint inequalities hold for commutators in Morrey spaces

Boundedness of classical harmonic analysis operators in Morrey spaces is established

Results apply to both weighted and unweighted cases

Abstract

Let $[b,\mathcal T_\alpha]~(0\leq\alpha<n)$ be the commutators generated by $BMO(\mathbb R^n)$ functions and a class of sublinear operators satisfying certain size conditions. The aim of this paper is to study the endpoint estimates of these commutators in the weighted Morrey spaces and in the generalized Morrey spaces, under the assumptions that $[b,\mathcal T_\alpha]~(0\leq\alpha<n)$ satisfy (weighted or unweighted) endpoint inequalities on $\mathbb R^n$ and on bounded domains. Furthermore, as applications of our main results, we will obtain, in the endpoint case, the boundedness properties of many important operators in classical harmonic analysis on the weighted Morrey and the generalized Morrey spaces.