Weyl's lemma on $RCD(K,N)$ metric measure spaces

TL;DR

This paper extends Weyl's lemma to $RCD(K,N)$ spaces, establishing local regularity, Liouville-type results, and gradient estimates for solutions of elliptic equations with discontinuous coefficients.

Contribution

It introduces a generalized Weyl's lemma for $RCD(K,N)$ spaces, enabling new regularity and Liouville results in this setting.

Findings

Proves local regularity of solutions to Poisson equations on $RCD(K,N)$ spaces.

Establishes Liouville-type theorem for $L^1$ weak harmonic functions.

Provides gradient estimates for elliptic equations with discontinuous coefficients.

Abstract

In this paper, we extend the classical Weyl's lemma to $RCD(K,N)$ metric measure spaces. As its applications, we show the local regularity of solutions for Poisson equations and a Liouville-type result for $L^1$ very weak harmonic functions on $RCD(K,N)$ spaces. Meanwhile, a byproduct is that we obtain a gradient estimate for solutions to a class of elliptic equations with dis-continuous coefficients.