Lipschitz continuity of solutions of Poisson equations in metric measure spaces

Abstract

Let $(X,d)$ be a pathwise connected metric space equipped with an Ahlfors $Q$-regular measure $\mu$, $Q\in[1,\infty)$. Suppose that $(X,d,\mu)$ supports a 2-Poincar\'e inequality and a Sobolev-Poincar\'e type inequality for the corresponding "Gaussian measure". The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation $\Delta u=f$, where $f\in L^p_\loc$. When $p>Q$, the local Lipschitz continuity of $u$ is established.