Quasibounded plurisubharmonic functions
Mårten Nilsson, Frank Wikström
TL;DR
This work develops a theory of quasibounded plurisubharmonic functions by extending the harmonic framework through the operator $S$ on functions admitting a plurisuperharmonic majorant, and applies Jensen measures together with Edwards' duality to Dirichlet problems with unbounded boundary data. It specializes to $\mathcal{PSH}$, yielding a Perron–Bremermann envelope $P\phi$ that is a maximal PSH function, continuous outside a pluripolar set, and unique when it is quasibounded; the results are tied to boundary data that are tame and to dual representations via Jensen measures. The approach links tameness, subextension, and envelope methods to establish existence, uniqueness, and boundary behavior for generalized Dirichlet problems, with implications for pluricomplex Green functions and unbounded boundary values in several complex variables. The paper thus provides robust tools for solving unbounded boundary-value problems in pluripotential theory and clarifies when unique maximal PSH solutions arise under tameness and quasiboundedness assumptions.
Abstract
We extend the notion of quasibounded harmonic functions to the plurisubharmonic setting. As an application, using the theory of Jensen measures, we show that certain generalized Dirichlet problems with unbounded boundary data admit unique solutions, and that these solutions are continuous outside a pluripolar set.