Characterizations of the mixed central Campanato spaces via the commutator operators of Hardy type

TL;DR

This paper characterizes mixed central Campanato spaces using the boundedness of Hardy type commutator operators, addressing technical challenges for negative lambda and introducing a mixed reverse H"older condition.

Contribution

It provides new characterizations of mixed central Campanato spaces via Hardy type commutators, including cases with negative lambda and additional reverse H"older assumptions.

Findings

Characterizations of mixed central Campanato spaces via commutator boundedness.

Overcoming technical difficulties for lambda<0.

Results on Hardy type operators on mixed lambda-central Morrey spaces.

Abstract

The purpose of this paper is to establish some characterizations of mixed central Campanato space $\mathfrak{C}^{\vec{p},\lambda}(\mathbb{R}^{n})$, via the boundedness of the commutator operators of Hardy type. Unlike the case $\lambda\geq0$, there are some technical difficulties caused by $\lambda<0$ to be overcome. In addition, an extra assumption called as the mixed version of the reverse H\"{o}lder class be required in the proof of the converse characterization. Moreover, some further interesting conclusions for the Hardy type operators on mixed $\lambda$-central Morrey space $\mathcal{B}^{\vec{p},\lambda}(\mathbb{R}^{n})$ are also derived.