Gradient Estimate for Solutions to Poisson Equations in Metric Measure Spaces

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 $\mathrm{\Delta }u=f$ 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.