Theorem 4.4.222B657EA5CC09

Proof. On $B(a,r)$ the function $u(y)=r-|y-a|$ peaks finitely at $a$. Nonthinness and the preceding theorem give $\widehat R_ Fu(a)=u(a)$. On the complement $w=u-\widehat R_ Fu$ is nonnegative superharmonic, tends to zero at $a$ by lower semicontinuity of the reduction, and is strictly positive: otherwise the strong minimum principle would make it zero on a component, contradicting $\Delta u=-(d-1)/|y-a|<0$ there.

Conversely, if $F$ were thin, its strict gap would give a subharmonic $v$ on a smaller ball with $v(a)=1$ and $v\le -1$ on $F\setminus \{ a\} $. On a still smaller concentric sphere, $v<0$ near $F$, and away from that neighborhood the positive lsc barrier $w$ has a positive lower bound. Choose $A$ so $v-Aw\le 0$ on that sphere. Subtracting a small positive fundamental pole at $a$ makes the boundary upper limit nonpositive at $a$ as well. The bounded-domain maximum principle gives $v\le Aw$ after the pole coefficient tends to zero. The upper limit of $v$ along the complement is at most zero and along $F$ at most $-1$, contradicting the canonical subharmonic value $v(a)=1$. The regularity equivalence now follows from the weak-barrier theorem on $\Omega \cap B(a,r)$ and the locality of finite boundary regularity. $\square$