The complex Sobolev Space and Hölder continuous solutions to Monge-Ampère equations

Abstract

Let $X$ be a compact Kähler manifold of dimension $n$ and $\omega$ a Kähler form on $X$. We consider the complex Monge-Ampère equation $(d{d}^{c}u+\omega {)}^n=\mu$, where $\mu$ is a given positive measure on $X$ of suitable mass and $u$ is an $\omega$-plurisubharmonic function. We show that the equation admits a Hölder continuous solution {\it if and only if} the measure $\mu$, seen as a functional on a complex Sobolev space $W^{\ast }(X)$, is Hölder continuous. A similar result is also obtained for the complex Monge-Ampère equations on domains of ${\mathbb{C}}^n$.

16 pages. Final version, to appear in Bulletin of the London Mathematical Society