Cheeger-harmonic functions in metric measure spaces revisited

Let (X, d, mu) be a complete metric measure space, with mu a locally doubling measure, that supports a local weak L-2-Poincare inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on (X, d, mu). Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented. (C) 2013 Elsevier Inc. All rights reserved.