Cheeger-harmonic functions in metric measure spaces revisited

摘要

Let (X, d, μ) be a complete metric measure space, with μ a locally doubling measure, that supports a local weak 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). Gradient estimates for Cheeger- harmonic functions and solutions to a class of non-linear Poisson type equations are presented.