New function classes of Morrey-Campanato type and their applications

The aim of this paper is to introduce and investigative some new function classes of Morrey-Campanato type. Let $0<p<\infty$ and $0\leq \lambda<n+p$. We say that $f\in \mathcal{\bar{L}}^{p,\lambda}(\Omega)$ if $$\sup_{x_{0}\in \Omega,\rho>0}\rho^{-\lambda}\int_{\Omega(x_{0},\rho)}\big|f(x)-|f|_{\Omega(x_{0},\rho)}\big|^pdx<\infty,$$ 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 $0\leq \lambda<n$, the space is equivalent to related Morrey space. If $\lambda=n$, then $f \in \mathcal{\bar{L}}^{p,n}(\Omega)$ if and only if $f\in BMO(\Omega)$ with $f^{-}\in L^{\infty}(\Omega)$, where $f^{-}=-\min\{0,f\}$. If $n<\lambda\leq n+p$, the $\mathcal{\bar{L}}^{p,\lambda}(\Omega)$ functions establish an integral characterization of the nonnegative H\"{o}lder continue functions. As applications, this paper gives unified criterions on the necessity of bounded commutators of maximal functions.