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