The Dirichlet problem for the Monge–Ampère equation on Hermitian manifolds with boundary

Abstract

We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge–Ampère equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and Hölder continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that are well dominated by capacity, for example measures with Lp, p>1 densities, or moderate measures in the sense of Dinh–Nguyen–Sibony.