Lemma 3.3.16B0CFE200676D

Proof. This is a short consequence of the checked local Riesz decomposition. On a sufficiently small ball define $u=h+\Phi (\cdot -a)$ off $a$ and $u(a)=+\infty $. The fundamental function tends to infinity, and $h\ge 0$; thus $u$ is lsc and is superharmonic, using locality off $a$ and the trivial supermean inequality at $a$. It is locally integrable by the checked local integrability theorem. Its positive Riesz measure $-\Delta u$ is supported at $a$, so it equals $b\delta _ a$ with $b\ge 0$. Since $-\Delta \Phi =c_ d\delta _ a$, where $c_2=2\pi $ and $c_ d=(d-2)s_{d-1}$ otherwise, Weyl's lemma gives $u=(b/c_ d)\Phi +v$ almost everywhere. Equality holds everywhere off $a$ by continuity. Hence $h=(b/c_ d-1)\Phi +v$ there, and its nonnegativity near $a$ forces $c=b/c_ d-1\ge 0$. This also proves removability when $h$ is bounded, or when a one-sided bounded harmonic function has a finite limit along one approach to $a$. $\square$