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

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

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