Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces

TL;DR

This paper establishes gradient estimates, regularity, and inequalities for solutions to Poisson equations on metric measure spaces with specific geometric and measure-theoretic properties.

Contribution

It introduces new gradient estimates and regularity results for Poisson equations in metric measure spaces supporting Poincaré inequalities and curvature bounds.

Findings

Established Moser-Trudinger and Sobolev inequalities for gradients

Proved local Hölder continuity of solutions with optimal exponent

Extended regularity theory to non-smooth metric measure spaces

Abstract

Let $(X,d)$ be a complete, pathwise connected metric measure space with locally Ahlfors $Q$-regular measure $\mu$, where $Q>1$. Suppose that $(X,d,\mu)$ supports a (local) $(1,2)$-Poincar\'e inequality and a suitable curvature lower bound. For the Poisson equation $\Delta u=f$ on $(X,d,\mu)$, Moser-Trudinger and Sobolev inequalities are established for the gradient of $u$. The local H\"older continuity with optimal exponent of solutions is obtained.