A Generalization of Exponential Class and Its Applications

Abstract

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