Characterization of Monge-Ampere measures with Holder continuous potentials

TL;DR

This paper characterizes when the complex Monge-Ampere equation on a compact Kaehler manifold admits a Holder continuous solution, linking it to the Holder continuity of the measure's super-potential and related function spaces.

Contribution

It establishes a precise equivalence between Holder continuous solutions and measures with Holder continuous super-potentials, extending to measures with potentials in various function spaces.

Findings

Holder continuous solutions exist iff the measure has Holder continuous super-potential

Measures with locally Holder continuous potentials satisfy the condition

Measures in certain Sobolev and Besov spaces also satisfy the condition

Abstract

We show that the complex Monge-Ampere equation on a compact Kaehler manifold (X,\omega) of dimension n admits a Holder continuous omega-psh solution if and only if its right-hand side is a positive measure with Holder continuous super-potential. This property is true in particular when the measure has locally Holder continuous potentials or when it belongs to the Sobolev space W^{2n/p-2+epsilon,p}(X) or to the Besov space B^{epsilon-2}_{\infty,\infty}(X) for some epsilon>0 and p>1.