Balayage of measures: behavior near a corner

TL;DR

The paper proves that the boundary behavior of balayage measures onto corners of opening $\pi\alpha$ is universal, with the vanishing rate of $\nu(\partial\Omega\cap B_r(z_0))$ as $r\to0$ determined solely by $\alpha$ and the density exponent $b$, yielding $r^{2b}$, $r^{2b}\log(1/r)$, or $r^{1/\alpha}$ depending on whether $2b<1/\alpha$, $2b=1/\alpha$, or $2b>1/\alpha$. It extends to multiple corners and provides explicit bounds when $2b\le 1/\alpha$, and it develops local geometric and harmonic-measure tools to establish decoupling and localization near corners. The results feed into applications to 2D Coulomb gases and hard-edge universality, linking boundary geometry to local microscopic statistics via the balayage measure. The work thus identifies universal corner-driven boundary behavior in potential theory with concrete constants and broad applicability.

Abstract

We consider the balayage of a measure $μ$ defined on a domain $Ω$ onto its boundary $\partial Ω$. Assuming that $Ω$ has a corner of opening $πα$ at a point $z_0 \in \partial Ω$ for some $0 < α\leq 2$ and that $dμ(z) \asymp |z-z_{0}|^{2b-2}d^{2}z$ as $z\to z_0$ for some $b > 0$, we obtain the precise rate of vanishing of the balayage of $μ$ near $z_{0}$. The rate of vanishing is universal in the sense that it only depends on $α$ and $b$. We also treat the case when the domain has multiple corners at the same point. Moreover, when $2b\leq \frac{1}α$, we provide explicit constants for the upper and lower bounds.

Figures (6)

• Figure 1: Illustration of a corner of opening $\pi \alpha$ at $z_0$.
• Figure 2: Illustration of an open set $\Omega$ with three corners at $z_0$.
• Figure 3: Illustration of Section \ref{['multiplestructuresubsec']} with $m=3$.
• Figure 4: Illustration of Lemma \ref{['extremaldistancelemma']}.
• Figure 5: Illustration of the argument leading to (\ref{['omegazEOmega']}) in the case of $|z| \geq \rho_0$.

Theorems & Definitions (20)

• Theorem 2.1: A single corner
• Remark 1
• Remark 2
• Remark 3
• Theorem 2.2: Multiple corners at the same point
• Lemma 3.1
• proof
• Lemma 4.1
• Lemma 4.2
• proof