H\"older continuous solutions of the Monge-Amp\`ere equation on compact Hermitian manifolds

TL;DR

This paper establishes a characterization for the existence of Holder continuous solutions to the Monge-Ampère equation on compact Hermitian manifolds, linking measure domination to solution regularity.

Contribution

It provides a necessary and sufficient condition for the existence of Holder continuous solutions to the Monge-Ampère equation in the Hermitian setting.

Findings

Holder continuous solutions exist if and only if the measure is locally dominated by measures of Holder continuous psh functions.

The result extends regularity theory of Monge-Ampère equations to Hermitian manifolds.

Characterization bridges measure domination and solution regularity in complex geometry.

Abstract

We show that a positive Borel measure of positive finite total mass, on compact Hermitian manifolds, admits a Holder continuous quasi-plurisubharmonic solution to the Monge-Ampere equation if and only if it is dominated locally by Monge-Ampere measures of Holder continuous plurisubharmonic functions.