Hölder continuous solutions to Monge-Ampère equations

Abstract

We study the regularity of solutions to complex Monge-Ampère equations $(d{d}^{c}u{)}^n=fdV$, on bounded strongly pseudoconvex domains $ Ω\subset \C^n$. We show, under a mild technical assumption, that the unique solution $u$ to such an equation is Hölder continuous if the boundary values $\varphi$ are Hölder continuous and the density $f$ belongs to $L^p(\Omega )$ for some $p>1$. This improves previous results by Bedford-Taylor and Kolodziej.