John--Nirenberg--Campanato 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 $J{N}_{(p,q,s{)}_{\alpha }}(\mathcal{X})$, where $\mathcal{X}$ is ${\mathbb{R}}^n$ or any closed cube $Q_0⫋{\mathbb{R}}^n$, which when $\alpha =0$ and $s=0$ coincides with the $J{N}_p$-space introduced by F. John and L. Nirenberg in the sense of equivalent norms. The authors then give the predual space of $J{N}_{(p,q,s{)}_{\alpha }}(\mathcal{X})$ 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 $J{N}_{(p,q,s{)}_{\alpha }}(\mathcal{X})$ when $p\to \infty$.

38 pages; submitted

Cited by in corpus (1)

• Localized John--Nirenberg--Campanato Spaces