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On the size of boundary pluripolar sets

Mårten Nilsson

TL;DR

The paper analyzes the size and propagation of boundary pluripolar sets, which are the exceptional sets for the Dirichlet problem for the complex Monge-Ampère equation. It extends Stout's results on peak sets to boundary b-pluripolar $F_\sigma$ sets, shows that such sets with topological dimension exceeding $N-1$ must propagate into the interior, and proves that sufficiently small Hausdorff dimension implies b-pluripolarity and non-propagation under suitable boundary regularity. It also proves that Jensen measures and representing measures do not coincide on smooth, strictly pseudoconvex domains, extending Hedenmalm, by constructing measures that separate the two classes. Together, these results clarify the boundary singularity structure and lifting properties of pluripotential-theoretic hulls and provide tools for analyzing solvability of the Dirichlet problem in several complex variables.

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--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.

On the size of boundary pluripolar sets

TL;DR

The paper analyzes the size and propagation of boundary pluripolar sets, which are the exceptional sets for the Dirichlet problem for the complex Monge-Ampère equation. It extends Stout's results on peak sets to boundary b-pluripolar sets, shows that such sets with topological dimension exceeding must propagate into the interior, and proves that sufficiently small Hausdorff dimension implies b-pluripolarity and non-propagation under suitable boundary regularity. It also proves that Jensen measures and representing measures do not coincide on smooth, strictly pseudoconvex domains, extending Hedenmalm, by constructing measures that separate the two classes. Together, these results clarify the boundary singularity structure and lifting properties of pluripotential-theoretic hulls and provide tools for analyzing solvability of the Dirichlet problem in several complex variables.

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--Ampère equation. We extend Stout's result that peak sets on strictly pseudoconvex domains must have topological dimension less than to also encompass non-propagating boundary pluripolar sets. In particular, boundary pluripolar sets must propagate into the interior if their topological dimension exceeds . 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.
Paper Structure (4 sections, 11 theorems, 45 equations, 1 figure)

This paper contains 4 sections, 11 theorems, 45 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Boundary pluripolar sets of dimension N must propagate
  3. Sets of small Hausdorff dimension are boundary pluripolar
  4. Jensen measures do not coincide with representing measures

Key Result

Lemma 2.1

Let $\Omega\subset \mathbb{C}^N$ be a bounded convex domain, and suppose that $A \subset \partial \Omega$ is a compact set such that for every neighborhood $U$ (in $\partial \Omega$) containing $p\in A$, there is a real hyperplane $\Pi_U: L(z)=0$ such that If each $(A\cap \{L(z)>0\}) \cup (\Pi_U \cap \widebar \Omega)$ is polynomially convex, then $\dim A < N$.

Figures (1)

  • Figure : Figure. A point $p\in \{L(z)>0\}\cap \partial \Omega \subset U$.

Theorems & Definitions (27)

  • Lemma 2.1: Stout
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark
  • Example 2.4
  • Theorem 2.5
  • proof
  • Remark
  • ...and 17 more