# A Generalization of Exponential Class and its Applications

## TL;DR

This paper introduces a new function space generalizing exponential classes, proves its Banach space properties, and applies it to analyze weak solutions of $a0\mathcal{A}$-harmonic equations, including boundedness of key operators.

## Contribution

It defines the space $L^{\theta,\infty)}(\Omega)$, proves its Banach space structure, and applies it to extend results on weak solutions and operator boundedness.

## Key findings

- $L^{\theta,\infty)}(\Omega)$ is a Banach space.
- Weak monotonicity property for $a0\mathcal{A}$-harmonic solutions is established.
- Boundedness of Hardy-Littlewood maximal and Calderf3n-Zygmund operators on the weighted space.

## Abstract

A function space, $L^{\theta,\infty)}(\Omega)$, $0 \leq \theta <\infty$, 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 $L^{\theta,\infty)}_w(\Omega)$) is also defined, and the boundedness for the Hardy-Littlewood maximal operator $M_w$ and a Calder\'{o}n-Zygmund operator $T$ with respect to $L^{\theta,\infty)}_w(\Omega)$ are obtained.