论文标题
复杂的Sobolev空间和Hölder的连续解决方案到Monge-ampère方程
The complex Sobolev Space and Hölder continuous solutions to Monge-Ampère equations
论文作者
论文摘要
令$ x $是$ n $ dimension $ n $的紧凑型kähler歧管和$ω$ akähler形式的$ x $。我们考虑复杂的monge-ampère方程$（dd^c u+ox）^n =μ$，其中$μ$是合适质量的$ x $的给定阳性，$ u $是$ω$ - plurisubharmonic函数。我们表明，该方程在hölder连续解决方案{\ it i时及时允许$μ$，被视为在复杂的sobolev space上的功能$ w^*（x）$，是hölder连续的。对于$ \ mathbb {c}^n $的域上的复杂的monge-ampère方程也获得了类似的结果。
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 $\mathbb{C}^n$.
