New John--Nirenberg--Campanato-Type Spaces Related to Both Maximal Functions and Their Commutators

TL;DR

This paper introduces a new class of function spaces of John-Nirenberg-Campanato type, characterizing their properties and relations to maximal functions and commutators, expanding the understanding of function space theory.

Contribution

The authors define a novel function space $ ilde{JN}_{(p,q,s)_{eta}}( ext{X})$ and establish its equivalence with existing spaces, including characterizations via maximal functions and commutators.

Findings

Established equivalent characterizations of the new space.

Proved basic properties and inequalities such as good-$lambda$ and John-Nirenberg inequalities.

Connected the new space to maximal functions and their commutators.

Abstract

Let $p,q\in [1,\infty]$, $\alpha\in{\mathbb{R}}$, and $s$ be a non-negative integer. In this article, the authors introduce a new function space $\widetilde{JN}_{(p,q,s)_{\alpha}}(\mathcal{X})$ of John-Nirenberg-Campanato type, where $\mathcal{X}$ denotes $\mathbb{R}^n$ or any cube $Q_{0}$ of $\mathbb{R}^n$ with finite edge length. The authors give an equivalent characterization of $\widetilde{JN}_{(p,q,s)_{\alpha}}(\mathcal{X})$ via both the John-Nirenberg-Campanato space and the Riesz-Morrey space. Moreover, for the particular case $s=0$, this new space can be equivalently characterized by both maximal functions and their commutators. Additionally, the authors give some basic properties, a good-$\lambda$ inequality, and a John-Nirenberg type inequality for $\widetilde{JN}_{(p,q,s)_{\alpha}}(\mathcal{X})$.