arXiv subjects

Mårten Nilsson

Publications and source records attributed to Mårten Nilsson.

6 recordsLinked to original sources

arXiv2026

On the size of boundary pluripolar sets

We prove a number of results related to the size and propagation of boundary pluripolar sets, the exceptional sets for the Dirichlet problem for the complex Monge--Ampère equation. We extend Stout's result that peak sets on strictly pseudoconvex domains $Ω\subset\mathbb{C}^N$ must have topological dimension less than $N$ to also encompass non-propagating boundary pluripolar $F_σ$ sets. In particular, boundary pluripolar sets must propagate into the interior if their topological dimension exceeds $N-1$. We also prove that sets of sufficiently small Hausdorff dimension must be boundary pluripolar and non-propagating, provided that the domain admits peak functions with sufficient boundary regularity. Lastly, we prove that the class of Jensen measures and the class of representing measures do not coincide on any smooth, strictly pseudoconvex domain. This extends a result of Hedenmalm.

math.CV↗

arXiv2025

Plurisubharmonic functions with discontinuous boundary behavior

We study the Dirichlet problem for the complex Monge-Ampère operator with bounded, discontinuous boundary data. If the set of discontinuities is b-pluripolar and the domain is B-regular, we are able to prove existence, uniqueness and some regularity estimates for a large class of complex Monge-Ampère measures. This result is optimal in the unit disk, as boundary functions with b-pluripolar discontinuity then coincides with functions that are continuous almost everywhere. We also show that neither of these properties of the boundary function - being continuous almost everywhere or having discontinuities forming a b-pluripolar set - are necessary conditions in order to establish uniqueness and continuity of the solution in higher dimensions. In particular, there are situations where it is enough to prescribe the boundary behavior at a set of arbitrarily small Lebesgue measure.

math.CV↗

arXiv2025

Quasibounded solutions to the complex Monge-Ampère equation

We study the Dirichlet problem for the complex Monge-Ampère operator on a B-regular domain $Ω$, allowing boundary data that is singular or unbounded. We introduce the concept of pluri-quasibounded functions on $Ω$ and $\partial Ω$, 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 $\mathcal{D}(Ω)$, 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.

math.CV↗

arXiv2023

Variations on a theorem by Edwards

We discuss two variations of Edwards' duality theorem. More precisely, we prove one version of the theorem for cones not necessarily containing all constant functions. In particular, we allow the functions in the cone to have a non-empty common zero set. In the second variation, we replace suprema of point evaluations and infima over Jensen measures by suprema of other continuous functionals and infima over a set measures defined through a natural order relation induced by the cone. As applications, we give some results on propagation of discontinuities for Perron--Bremermann envelopes in hyperconvex domains as well as a characterization of minimal elements in the order relation mentioned above.

math.CV↗

arXiv2021

Continuity of envelopes of unbounded plurisubharmonic functions

On bounded B-regular domains, we study envelopes of plurisubharmonic functions bounded from above by a function $ϕ$ such that $ϕ^*=ϕ_*$ on the closure of the domain. For $ϕ$ satisfying certain additional criteria limiting its behavior at the singularities, we establish a set where the Perron-Bremermann envelope $P ϕ$ is guaranteed to be continuous. This result is a generalization of a classic result in pluripotential theory due to J. B. Walsh. As an application, we show that the complex Monge--Ampère equation \[ (dd^cu)^n = μ\] being uniquely solvable for continuous boundary data implies that it is also uniquely solvable for a class of boundary values unbounded from above.

math.CV↗

arXiv2020

Quasibounded plurisubharmonic functions

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.

math.CV↗