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The aim of this paper is to introduce and investigative some new functionclasses of Morrey-Campanato type. Let $0<p<\infty$ and $0\leq \lambda<n+p$. Wesay 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 denotethe cube of $\mathbb{R}^n$. Some basic properties and characterizations ofthese classes are presented. If $0\leq \lambda<n$, the space is equivalent torelated 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 $n<\lambda\leqn+p$, the $\mathcal{\bar{L}}^{p,\lambda}(\Omega)$ functions establish anintegral characterization of the nonnegative H\"{o}lder continue functions. Asapplications, this paper gives unified criterions on the necessity of boundedcommutators of maximal functions.

New function classes of Morrey-Campanato type and their applications