H\"older continuous solutions to Monge-Amp\`ere equations

Abstract

We study the regularity of solutions to complex Monge-Amp\`ere equations $(dd^c u)^n=f dV$, on bounded strongly pseudoconvex domains $ \Omega \subset \C^n$. We show, under a mild technical assumption, that the unique solution $u$ to such an equation is H\"older continuous if the boundary values $\phi$ are H\"older continuous and the density $f$ belongs to $L^p(\Omega)$ for some $p>1$. This improves previous results by Bedford-Taylor and Kolodziej.