Localized John--Nirenberg--Campanato Spaces

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 $Q_0⫋{\mathbb{R}}^n$. In this article, the authors first introduce the localized John--Nirenberg--Campanato space $j{n}_{(p,q,s{)}_{\alpha }}(\mathcal{X})$ and show that the localized Campanato space is the limit case of $j{n}_{(p,q,s{)}_{\alpha }}(\mathcal{X})$ as $p\to \infty$. By means of local atoms and the weak-$\ast$ topology, the authors then introduce the localized Hardy-kind space $h{k}_{(p^{\prime },q^{\prime },s{)}_{\alpha }}(\mathcal{X})$ which proves the predual space of $j{n}_{(p,q,s{)}_{\alpha }}(\mathcal{X})$. Moreover, the authors prove that $h{k}_{(p^{\prime },q^{\prime },s{)}_{\alpha }}(\mathcal{X})$ is invariant when $1<q<p$, where $p^{\prime }$ or $q^{\prime }$ denotes the conjugate number of $p$ or $q$, respectively. All these results are new even for the localized John--Nirenberg space.

38 pages; Submitted

References in corpus (1)

• John--Nirenberg--Campanato Spaces