Balayage of measures: behavior near a corner

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.