On the size of boundary pluripolar sets

TL;DR

This paper investigates the size and propagation properties of boundary pluripolar sets in complex analysis, extending existing results and establishing new conditions under which these sets are small or propagate into the interior.

Contribution

It extends Stout's result to non-propagating boundary pluripolar sets, characterizes small Hausdorff dimension sets as boundary pluripolar, and shows Jensen and representing measures differ on smooth pseudoconvex domains.

Findings

Boundary pluripolar sets with topological dimension > N-1 propagate into the interior.

Sets with small Hausdorff dimension are boundary pluripolar and non-propagating.

Jensen measures and representing measures do not coincide on smooth strictly pseudoconvex domains.

Abstract

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--Amp\`ere equation. We extend Stout's result that peak sets on strictly pseudoconvex domains $\Omega\subset\mathbb{C}^N$ must have topological dimension less than $N$ to also encompass non-propagating boundary pluripolar $F_\sigma$ 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.