Quasibounded solutions to the complex Monge-Amp\`ere equation

TL;DR

This paper investigates the Dirichlet problem for the complex Monge-Ampère equation on B-regular domains, introducing pluri-quasibounded functions to handle singular boundary data, and establishes existence, uniqueness, and boundary behavior of solutions.

Contribution

It introduces the concept of pluri-quasibounded functions, extending solution existence and uniqueness to singular boundary data in complex Monge-Ampère problems.

Findings

Existence and uniqueness of solutions in the Blocki--Cegrell class for singular boundary data.

Extension of harmonic function representation to all regular domains in ^n.

Description of boundary singularity propagation into the interior.

Abstract

We study the Dirichlet problem for the complex Monge-Amp\`ere operator on a B-regular domain $\Omega$, allowing boundary data that is singular or unbounded. We introduce the concept of pluri-quasibounded functions on $\Omega$ and $\partial \Omega$, defined by the existence of plurisuperharmonic majorants that dominate their absolute value in a strong sense - that is, the ratio of the function to the majorant tends to zero as the function tends to infinity. For such data, we prove existence and uniqueness of solutions in the Blocki--Cegrell class $\mathcal{D}(\Omega)$, using a recently established comparison principle. In the unit disk, our approach recovers harmonic functions represented as Poisson integrals of $L^1$ boundary data with respect to harmonic measure, and our characterization extends to all regular domains in $\mathbb{R}^n$, when the boundary data is continuous almost everywhere. We also describe how boundary singularities propagate into the interior via a refined pluripolar hull.