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.

💡 Research Summary

This paper investigates the size and propagation properties of boundary pluripolar sets—exceptional sets for the Dirichlet problem of the complex Monge–Ampère equation—on bounded domains in ℂⁿ. The authors extend several classical results concerning peak sets to the more general class of boundary pluripolar (b‑pluripolar) Fσ sets.

Theorem A (Theorem 2.5).
Let Ω⊂ℂⁿ be strictly pseudoconvex. If a boundary pluripolar Fσ set A has topological dimension ≥ n, then its pluripolar hull ˆA meets the interior of Ω; in other words, A must propagate into Ω. Consequently, any non‑propagating b‑pluripolar set can have topological dimension at most n − 1. The proof adapts Stout’s argument for peak sets: assuming non‑propagation, Lemma 2.3 shows that for every real hyperplane Π intersecting a neighbourhood of a point of A, the union (A∩{L>0})∪(Π∩∂Ω) is polynomially convex. Lemma 2.1 (Stout’s cohomological lemma) then forces dim A < n, yielding a contradiction. The argument relies on B‑regularity (existence of strong plurisubharmonic barriers) and on approximation results (Wikström, Oka–Weil, Bremermann–Sibony) to produce the required plurisubharmonic functions.

Theorem B (Theorem 3.2).
Define P_β⊂∂Ω as the set of boundary points ζ for which there exists a peak function h_ζ∈H^∞(Ω) satisfying |h_ζ(z)−1|≤C_ζ‖z−ζ‖^β. If A⊂P_β has zero β‑dimensional Hausdorff measure (H_β(A)=0), then A is a boundary pluripolar, non‑propagating set. The construction uses the harmonic measure of small arcs on the unit disc (Lemma 3.3) to build plurisubharmonic functions u_{j,n}(z)=−ω(h_ζ(z),…) that are uniformly bounded away from −∞ at a fixed interior point a, while tending to −∞ along A. By taking a suitable series of such functions and exploiting the β‑Hölder control of the peak functions, the authors obtain a global psh function u with lim sup_{w→A}u(w)=−∞ and u(a)>−∞, establishing b‑pluripolarity and non‑propagation. The proof again uses approximation theorems to replace the constructed functions by logarithms of polynomials.

Theorem C (Corollary 4.4).
On any smooth strictly pseudoconvex domain Ω⊂ℂⁿ (n>1), the class of Jensen measures J_{z₀}(∂Ω) at a point z₀∈Ω does not coincide with the class of representing measures M_{z₀}(∂Ω). The authors construct a non‑propagating b‑pluripolar set K⊂∂Ω and a plurisubharmonic function u that blows up on K but remains finite at z₀. The associated representing measure μ supported on K fails the Jensen inequality log|f(z₀)|≤∫{∂Ω}log|f| dμ for all f∈A(Ω), showing μ∉J{z₀}(∂Ω). This generalizes Hedenmalm’s result for the unit ball to arbitrary smooth strictly pseudoconvex domains.

Additional contributions and remarks.

• Proposition 2.6 provides explicit polynomial constructions on the unit ball that realize any prescribed boundary set A and interior point z₀∉ˆA via sums of logarithms of polynomials.
• Example 2.4 shows that polynomial convexity alone does not prevent propagation: an arc on the sphere of the unit ball is b‑pluripolar and polynomially convex but propagates along a complex line.
• The paper discusses the sharpness of the dimension thresholds: while dim A≥n forces propagation, there exist non‑propagating b‑pluripolar sets of Hausdorff dimension up to 2n−1 (peak sets of maximal possible size).
• Remarks connect the results to known potential‑theoretic thresholds (Labutin’s logarithmic Hausdorff condition) and to constructions of non‑peakable boundary points (Y. Yu’s example).

Methodological tools.
The work combines several deep techniques:

1. Approximation of psh functions on B‑regular domains (Wikström).
2. Oka–Weil theorem to replace holomorphic functions by polynomials.
3. Bremermann–Sibony representation of psh functions as maxima of logarithms of holomorphic functions.
4. Harmonic measure estimates for arcs (Lemma 3.3) to control the size of constructed psh barriers.
5. Cohomological arguments (Stout) to relate polynomial convexity and topological dimension.

Overall, the paper provides a comprehensive quantitative description of when a boundary set can be “invisible” to the Monge–Ampère Dirichlet problem. By linking topological dimension, Hausdorff dimension, and regularity of peak functions, it clarifies the delicate balance between size and propagation for boundary pluripolar sets, and it settles a natural question about the distinction between Jensen and representing measures in several complex variables.