Complex Monge-Amp\`ere equations for plurifinely plurisubharmonic functions

TL;DR

This paper investigates complex Monge-Ampère equations within the framework of plurifinely plurisubharmonic functions, providing conditions for solutions involving measures with singular parts in hyperconvex domains.

Contribution

It introduces new conditions under which complex Monge-Ampère equations can be solved for measures with singular parts in the plurifinely setting.

Findings

Established sufficient conditions for solving Monge-Ampère equations with singular measures.

Extended the theory of plurisubharmonic functions to the plurifinely context.

Provided insights into the structure of solutions in hyperconvex domains.

Abstract

This paper studies the complex Monge-Amp\`ere equations for $\mathcal F$-plurisubharmonic functions in bounded $\mathcal F$-hyperconvex domains. We give sufficient conditions for this equation to solve for measures with a singular part.