Weakly solutions to the complex Monge-Amp\`ere equation on bounded plurifinely hyperconvex domains

TL;DR

This paper investigates conditions under which a plurifinely plurisubharmonic function exists that solves a complex Monge-Ampère equation with a given measure on bounded hyperconvex domains, expanding understanding of weak solutions.

Contribution

It provides sufficient conditions for the existence of weak solutions to the complex Monge-Ampère equation on bounded plurifinely hyperconvex domains.

Findings

Established criteria for measure μ to admit a plurifinely plurisubharmonic solution.

Extended the theory of complex Monge-Ampère equations to plurifinely hyperconvex domains.

Identified conditions ensuring solutions in the plurifine topology.

Abstract

Let $\mu$ be a non-negative measure defined on bounded $\mathcal F$-hyperconvex domain $\Omega$. We are interested in giving sufficient conditions on $\mu$ such that we can find a plurifinely plurisubharmonic function satisfying $NP (dd^c u)^n =\mu$ in $QB(\Omega)$.