Cheeger-harmonic functions in metric measure spaces revisited

TL;DR

This paper proves that Cheeger-harmonic functions are Lipschitz continuous in certain metric measure spaces under a heat semigroup curvature condition, providing gradient estimates and solutions to nonlinear equations.

Contribution

It establishes Lipschitz regularity of Cheeger-harmonic functions in metric measure spaces with a heat semigroup curvature condition, extending previous results.

Findings

Cheeger-harmonic functions are Lipschitz continuous under the given conditions.

Gradient estimates for Cheeger-harmonic functions are derived.

Solutions to nonlinear Poisson equations are analyzed.

Abstract

Let $(X,d,\mu)$ be a complete metric measure space, with $\mu$ a locally doubling measure, that supports a local weak $L^2$-Poincar\'e 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.