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, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $\mathrm{MAH}(X,\omega )$ 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 $\mathrm{MAH}(X,\omega )$ and proving that $\mathrm{MAH}X,\omega )$ has the “$L^p$-property”, $p>1$. We also describe accurately the symmetric measures it contains.