# On the H\"older continuous subsolution problem for the complex Monge-Amp\`ere equation

## TL;DR

This paper characterizes when the complex Monge-Ampère equation with zero boundary data has Hölder continuous solutions, providing a complete criterion based on the measure and addressing a question by Zeriahi.

## Contribution

It establishes a necessary and sufficient condition for measures to admit Hölder continuous solutions to the Dirichlet problem for the complex Monge-Ampère equation.

## Key findings

- Provides a full characterization of measures for Hölder solutions
- Answers Zeriahi's question affirmatively in the finite mass case
- Advances understanding of regularity in complex Monge-Ampère equations

## Abstract

We give a necessary and sufficient condition for positive Borel measures such that the Dirichlet problem, with zero boundary data, for the complex Monge-Amp\`ere equation admits H\"older continuous plurisubharmonic solutions. In particular, when the subsolution has finite Monge-Amp\`ere total mass, we obtain an affirmative answer to a question of Zeriahi.