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

TL;DR

This paper solves the Dirichlet problem for the complex Monge-Ampère equation on strictly pseudoconvex domains with measures dominated by H"older continuous plurisubharmonic functions, establishing continuity and H"older continuity of solutions.

Contribution

It provides existence and regularity results for solutions to the complex Monge-Ampère equation under new measure domination conditions, answering a question by Zeriahi.

Findings

Solutions are continuous if boundary data is continuous.

Solutions are H"older continuous if boundary is H"older continuous.

The results affirmatively answer Zeriahi's question.

Abstract

We solve the Dirichlet problem for the complex Monge-Amp\`ere equation on a strictly pseudoconvex with the right hand side being a positive Borel measure which is dominated by the Monge-Amp\`ere measure of a H\"older continuous plurisubharmonic function. If the boundary data is continuous, then the solution is continuous. If the boundary is H\"older continuous, then the solution is also H\"older continuous. In particular, the answer to a question of A. Zeriahi is always affirmative.