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. Let $0<p<\infty$ and $0\le \lambda <n+p$. We say that $f\in {\bar{\mathcal{L}}}^{p,\lambda }(\Omega )$ if $$\underset{x_0\in \Omega ,\rho >0}{\sup }{\rho }^{-\lambda }{\int }_{\Omega (x_0,\rho )}|f(x)-|f{|}_{\Omega (x_0,\rho )}{|}^{p}dx<\infty ,$$ 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. If $\lambda =n$, then $f\in {\bar{\mathcal{L}}}^{p,n}(\Omega )$ if and only if $f\in BMO(\Omega )$ 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 criterions on the necessity of bounded commutators of maximal functions.

29 pages