Characterization of BMO spaces via commutators of fractional maximal function on Lorentz spaces

TL;DR

This paper characterizes BMO spaces through the boundedness of commutators of fractional maximal functions on Lorentz spaces, providing new insights into the structure of BMO spaces.

Contribution

It introduces necessary and sufficient conditions for the boundedness of these commutators on Lorentz spaces, offering novel characterizations of BMO subclasses.

Findings

Characterization of BMO spaces via commutators

Necessary and sufficient boundedness conditions established

New subclasses of BMO spaces characterized

Abstract

Let $0 \leq \alpha<n$ and $b$ be the locally integrable function. In this paper, we consider the maximal commutator of fractional maximal function $M_{b,\alpha}$ and the nonlinear commutator of fractional maximal function $[b, M_{\alpha}]$ on Lorentz spaces. We give some necessary and sufficient conditions for the boundedness of the commutators $M_{b,\alpha}$ and $[b, M_{\alpha}]$ on Lorentz spaces when the function $b$ belongs to BMO spaces, by which some new characterizations of certain subclasses of BMO spaces are obtained.