# Localized John--Nirenberg--Campanato Spaces

## TL;DR

This paper introduces localized John--Nirenberg--Campanato spaces and their preduals, establishing new relationships and invariance properties, thereby extending the theory of localized function spaces.

## Contribution

The authors define new localized function spaces, connect them via duality, and prove invariance properties, advancing the understanding of localized harmonic analysis.

## Key findings

- Defined localized John--Nirenberg--Campanato spaces.
- Established the predual relationship with localized Hardy-kind spaces.
- Proved invariance of the Hardy-kind space under certain parameter conditions.

## Abstract

Let $p\in(1,\infty)$, $q\in[1,\infty)$, $s\in{\mathbb Z}_{+}$, $\alpha\in[0,\infty)$ and $\mathcal{X}$ be $\mathbb R^n$ or a cube $Q_0\subsetneqq\mathbb R^n$. In this article, the authors first introduce the localized John--Nirenberg--Campanato space $jn_{(p,q,s)_{\alpha}}(\mathcal{X})$ and show that the localized Campanato space is the limit case of $jn_{(p,q,s)_{\alpha}}(\mathcal{X})$ as $p\to\infty$. By means of local atoms and the weak-$*$ topology, the authors then introduce the localized Hardy-kind space $hk_{(p',q',s)_{\alpha}}(\mathcal{X})$ which proves the predual space of $jn_{(p,q,s)_{\alpha}}(\mathcal{X})$. Moreover, the authors prove that $hk_{(p',q',s)_{\alpha}}(\mathcal{X})$ is invariant when $1<q<p$, where $p'$ or $q'$ denotes the conjugate number of $p$ or $q$, respectively. All these results are new even for the localized John--Nirenberg space.