  Let $X$ be a compact K\"ahler manifold of dimension $n$ and $\omega$ a
K\"ahler form on $X$. We consider the complex Monge-Amp\`ere equation $(dd^c
u+\omega)^n=\mu$, 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 H\"older continuous solution {\it if and only if} the measure
$\mu$, seen as a functional on a complex Sobolev space $W^*(X)$, is H\"older
continuous. A similar result is also obtained for the complex Monge-Amp\`ere
equations on domains of $\mathbb{C}^n$.
