A necessary and sufficient condition for H\"{o}lder-class solutions to the complex Monge--Amp\`{e}re equation

TL;DR

This paper establishes a precise criterion for the existence of H"{o}lder continuous solutions to the complex Monge--Amp ext{e}re equation on bounded domains in complex space, advancing understanding of solution regularity.

Contribution

It provides a necessary and sufficient condition for H"{o}lder solutions and explores regularity on general B-regular domains, building on prior work by S.-Y. Li.

Findings

Characterization of H"{o}lder solutions criteria

Regularity results on B-regular domains

Extension of previous theoretical frameworks

Abstract

We provide a necessary and sufficient condition for the existence of H\"{o}lder continuous solutions to the complex Monge--Amp\`{e}re equation on bounded domains in $\mathbb{C}^n$. This condition is motivated by a paper by S.-Y. Li. We also prove a result on the regularity of solutions to the complex Monge--Amp\`{e}re equation on general $B$-regular domains in $\mathbb{C}^n$.