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 μ, Q ∈ [1, ∞ ). Suppose that (X, d, μ) supports a 2-Poincaré inequality and a Sobolev–Poincaré type inequality for the corresponding “Gaussian measure”. The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation Δu = f, where \(f\in L^{p}_{\rm{loc}}\). When p > Q, the local Lipschitz continuity of u is established.