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

Let $p,q\in [1,\mathrm{\infty }]$, $\alpha \in \mathbb{R}$, 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 ${\widetilde{JN}}_{(p,q,s{)}_{\alpha }}(\mathcal{X})$.