H\"older continuous solutions to Monge-Amp\`ere equations

TL;DR

This paper establishes uniform Hölder regularity for solutions to the complex Monge-Ampère equation on compact Kähler manifolds with $L^p$ data, and characterizes the range of the Monge-Ampère operator on Hölder continuous functions.

Contribution

It proves Hölder regularity for solutions with $L^p$ right-hand side and characterizes the operator's range, extending Ko{\

Findings

Hölder regularity for solutions with $L^p$ data.

Convexity and $L^p$-property of the Monge-Ampère operator's range.

Precise description of symmetric measures in the operator's range.

Abstract

Let $(X,\omega)$ be a compact K\"ahler manifold. We obtain uniform H\"older regularity for solutions to the complex Monge-Amp\`ere 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 $\MAH(X,\omega)$ of the complex Monge-Amp\`ere operator acting on $\omega$-plurisubharmonic H\"older continuous functions. We show that this set is convex, by sharpening 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.