Hölder regularity for solutions to complex Monge-Ampère equations

Abstract

We consider the Dirichlet problem for the complex Monge-Ampère equation in a bounded strongly hyperconvex Lipschitz domain in $\C^n$. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is continuous and the right hand side has a continuous density. Then we consider the case when the boundary value function is $C^{1,1}$ and the right hand side has a density in $L^p(\Omega )$ for some $p>1$ and prove the Hölder continuity of the solution.