The complex Sobolev space and Hölder continuous solutions to Monge–Ampère equations

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