New function classes of Morrey–Campanato type and their applications

Abstract

The aim of this paper is to introduce and investigative some new function classes of Morrey–Campanato type. The domains \(\Omega \subset {\mathbb {R}}^n\) in this paper are supposed to satisfy the following property: there exists a constant \(A>0\) such that for all \(x_{0}\in \Omega , \rho <\mathrm{diam} \ \Omega \), we have \(|Q(x_{0},\rho )\cap \Omega |\ge A\rho ^n.\) Let \(0<p<\infty \) and \(0\le \lambda \le n+p\). We say that \(f\in \bar {\mathcal{L}}^{p,\lambda }(\Omega )\) if $$\begin{aligned} \sup _{x_{0}\in \Omega ,\rho >0}\rho ^{-\lambda }\int _{\Omega (x_{0},\rho )}\big |f(x)-|f|_{\Omega (x_{0},\rho )}\big |^p {\mathrm{d}}x<\infty , \end{aligned}$$where \(\Omega (x_{0},\rho )=Q(x_{0},\rho )\cap \Omega \) and \(Q(x,\rho )\) is denote the cube of \({\mathbb {R}}^n\). Some basic properties and characterizations of these classes are presented. If \(0\le \lambda <n\), the space is equivalent to related Morrey space under certain conditions. If \(\lambda =n\), then \(f \in \bar {\mathcal{L}}^{p,n}(\Omega )\) if and only if \(f\in {\mathrm{BMO}}(\Omega )\) with \(f^{-}\in L^{\infty }(\Omega )\), where \(f^{-}=-\min \{0,f\}\). If \(n<\lambda \le n+p\), the \(\bar {\mathcal{L}}^{p,\lambda }(\Omega )\) functions establish an integral characterization of the nonnegative Hölder continue functions. As applications, this paper gives unified criteria on the necessity of bounded commutators of maximal functions on ball Banach function spaces.