Hölder continuous solutions to Monge–Ampère equations

Abstract

Let $(X,\omega)$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, \hbox{$p>1$}. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $\MAH(X,\omega)$ of the complex Monge-Ampère operator acting on $\omega$-pluri\\-subharmonic Hölder continuous functions. We show that this set is convex, by sharpening\break Ko\l odziej's result that measures with $L^p$-density belong to $\MAH(X,\omega)$ and proving that $\MAH(X,\omega)$ has the "$L^p$-property'', $p>1$. We also describe accurately the symmetric measures it contains.