John--Nirenberg--Campanato Spaces

TL;DR

This paper introduces a new class of function spaces called John--Nirenberg--Campanato spaces, explores their properties, dual spaces, and relationships to classical Campanato spaces, extending the understanding of function space theory.

Contribution

The authors define the John--Nirenberg--Campanato spaces, establish their predual spaces, prove a John--Nirenberg inequality, and connect these spaces to classical Campanato spaces as a limit case.

Findings

Defined the John--Nirenberg--Campanato spaces $JN_{(p,q,s)_eta}( ext{X})$.

Established the predual space of $JN_{(p,q,s)_eta}( ext{X})$.

Proved a John--Nirenberg type inequality for these spaces.

Abstract

Let $p\in (1,\infty)$, $q\in[1,\infty)$, $\alpha\in [0,\infty)$ and $s$ be a non-negative integer. In this article, the authors introduce the John--Nirenberg-Campanato space $JN_{(p,q,s)_\alpha}(\mathcal{X})$, where $\mathcal{X}$ is ${\mathbb R}^n$ or any closed cube $Q_0\subsetneqq{\mathbb R}^n$, which when $\alpha=0$ and $s=0$ coincides with the $JN_p$-space introduced by F. John and L. Nirenberg in the sense of equivalent norms. The authors then give the predual space of $JN_{(p,q,s)_\alpha}(\mathcal{X})$ and a John-Nirenberg type inequality of John--Nirenberg-Campanato spaces. Moreover, the authors prove that the classical Campanato space serves as a limit space of $JN_{(p,q,s)_\alpha}(\mathcal{X})$ when $p\to \infty$.