Cheeger-harmonic functions in metric measure spaces revisited

Let (X,d,μ)$(X,d,\mu )$ be a complete metric measure space, with μ   a locally doubling measure, that supports a local weak L²$L^2$-Poincaré inequality. By assuming a heat semigroup type curvature condition, we prove that Cheeger-harmonic functions are Lipschitz continuous on (X,d,μ)$(X,d,\mu )$. Gradient estimates for Cheeger-harmonic functions and solutions to a class of non-linear Poisson type equations are presented.