Characterizations of the BMO and Lipschitz spaces via commutators on weak Lebesgue and Morrey spaces

TL;DR

This paper characterizes BMO and Lipschitz spaces through commutator boundedness on weak Lebesgue and Morrey spaces, establishing new inclusion relations and equivalences for Calderón-Zygmund operators.

Contribution

It introduces novel characterizations of BMO and Lipschitz spaces via commutator boundedness on weak Morrey and Morrey spaces, extending previous results.

Findings

Weak Morrey space $WM^{p}_{q}$ is contained in $M^{p}_{q_{1}}$ for certain parameters.

Boundedness of commutators characterizes BMO functions.

Similar results are obtained for Lipschitz functions.

Abstract

We prove that the weak Morrey space $WM^{p}_{q}$ is contained in the Morrey space $M^{p}_{q_{1}}$ for $1\leq q_{1}< q\leq p<\infty$. As applications, we show that if the commutator $[b,T]$ is bounded from $L^p$ to $L^{p,\infty}$ for some $p\in (1,\infty)$, then $b\in \mathrm{BMO}$, where $T$ is a Calder\'on-Zygmund operator. Also, for $1<p\leq q<\infty$, $b\in \mathrm{BMO}$ if and only if $[b,T]$ is bounded from $M^{p}_{q}$ to $WM_{q}^{p}$. For $b$ belonging to Lipschitz class, we obtain similar results.