Quasibounded solutions to the complex Monge–Ampère equation

Abstract We study the Dirichlet problem for the complex Monge–Ampère operator on B‐regular domains in , allowing boundary data that is singular or unbounded. We extend the concept of pluri‐quasibounded functions on the domain to functions on the boundary, 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 boundary data, we prove existence and uniqueness of pluri‐quasibounded solutions in the Błocki–Cegrell class, the largest class for which the complex Monge–Ampère operator is well‐behaved. In the unit disk, our approach recovers harmonic functions represented as Poisson integrals of boundary data with respect to harmonic measure, and our characterization extends to all regular domains in , when the boundary data is continuous almost everywhere. We also describe how boundary singularities propagate into the interior via a refined pluripolar hull.