Lemma 4.4.1358BE7FB3C948

Proof. On compact subdomains avoiding the center, transform harmonic comparison functions, use positivity of the Kelvin multiplier and its involutivity, and transform the comparison back. This proves the superharmonic comparison property for the transformed lsc function. In the normalized case the transform is harmonic near infinity with zero large-sphere means: the original function is harmonic near $a$ with zero center value, so the concentric mean identity and the flux transfer in the exterior theorem apply. In particular its value at infinity is zero and it is continuous there. For any other exterior ball in the domain, its sphere means are concave in $r^{2-d}$ (or $-\log r$ in the plane), and are zero for all sufficiently large radii. Concavity makes them nonpositive at the smaller allowed radii. This is exactly the supermean condition at infinity. $\square$