Lipschitz continuity of solutions of Poisson equations in metric measure spaces

Let $(X,d)$ be a pathwise connected metric space equipped with an Ahlfors $Q$-regular measure $\mu$, $Q\in [1,\mathrm{\infty })$. 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 $\mathrm{\Delta }u=f$, where $f\in L_{\text{loc}}^p$. When $p>Q$, the local Lipschitz continuity of $u$ is established.