Bourgain-Morrey-Lorentz spaces and operators on them

TL;DR

This paper introduces Bourgain-Morrey-Lorentz spaces, explores their duality and operator boundedness, and characterizes BMO functions via commutator boundedness and compactness within these spaces.

Contribution

It defines Bourgain-Morrey-Lorentz spaces, establishes duality with block spaces, and analyzes boundedness and compactness of classical operators on these spaces.

Findings

Boundedness of Hardy-Littlewood maximal, sharp maximal, Calderón-Zygmund, and fractional integral operators.

Characterization of BMO functions via commutator boundedness.

Compactness of commutators characterized by limits of smooth functions in BMO.

Abstract

We introduce Bourgain-Morrey-Lorentz spaces and give a description of the predual of Bourgain-Morrey-Lorentz spaces via the block spaces. As an application of duality, we obtain the boundedness of Hardy-Littlewood maximal operator, sharp maximal operator, Calder\'on-Zygmund operator, fractional integral operator, commutator on Bourgain-Morrey-Lorentz spaces. Moreover, we obtain a weak Hardy factorization terms of Calder\'on-Zygmund operator in Bourgain-Morrey-Lorentz spaces. Using this result, we obtain a characterization of functions in $\BMO$ (the functions of ``bounded mean oscillation'') via the boundedness of commutators generated by them and a homogeneous Calder\'on-Zygmund operator. In the last, we show that the commutator generated by a function $b$ and a homogeneous Calder\'on-Zygmund operator is a compact operator on Bourgain-Morrey-Lorentz spaces if and only if $b$ is the limit of compactly supported smooth functions in $\BMO$.