Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces

TL;DR

This paper extends the sharp maximal theorem to mixed-norm Lebesgue spaces and characterizes BMO and Lipschitz spaces through the boundedness of commutators of fractional integral operators, providing new insights into their boundedness conditions.

Contribution

It generalizes the sharp maximal theorem to mixed-norm spaces and characterizes BMO and Lipschitz spaces via commutator boundedness, offering new theoretical tools.

Findings

Characterization of BMO via commutator boundedness

Characterization of Lipschitz space via commutator boundedness

Applications of the main corollary to specific operator bounds

Abstract

In this paper, the sharp maximal theorem is generalized to mixed-norm ball Banach function spaces, which is defined as Definition 2.7. As an application, we give a characterization of BMO via the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Moreover, the characterization of homogeneous Lipschitz space is also given by the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Finally, two applications of Corollary 6.4 are given.