A Generalization of Exponential Class and its Applications

Abstract

A function space, $L^{\theta ,\infty )}(\Omega )$, $0\le \theta <\infty$, is defined. It is proved that $L^{\theta ,\infty )}(\Omega )$ is a Banach space which is a generalization of exponential class. An alternative definition of $L^{\theta ,\infty )}(\Omega )$ space is given. As an application, we obtain weak monotonicity property for very weak solutions of $\mathcal{A}$-harmonic equation with variable coefficients under some suitable conditions related to $L^{\theta ,\infty )}(\Omega )$, which provides a generalization of a known result due to Moscariello. A weighted space $L_w^{\theta ,\infty )}(\Omega )$) is also defined, and the boundedness for the Hardy-Littlewood maximal operator $M_w$ and a Calderón-Zygmund operator $T$ with respect to $L_w^{\theta ,\infty )}(\Omega )$ are obtained.