Lemma 4.4.15E0D6C3CD3750

Proof. Choose a closed ball disjoint from $\overline\Omega $ and invert about its sphere, sending $\Omega $ into a bounded ball. Continuous data, after subtracting a constant and a uniform approximation if necessary, can be assumed zero near infinity. The transformed data, divided by the positive Kelvin multiplier, then extend continuously with value zero near the inversion center. If infinity is interior, the center is separated from the transformed boundary and the same prescription is possible. Approximate uniformly by smooth functions $\varphi $ zero near that center. Such smooth approximation follows by extending a continuous function from the compact boundary, multiplying by a cutoff, and convolving; the uniform error on the boundary is controlled by uniform continuity.

These approximants are differences of superharmonic functions harmonic and zero near the center. To verify this, take $q(x)=-A(|x-a|-r_0)_+^4$, with the support of $\Delta \varphi $ separated from $\overline{B(a,r_0)}$. Its Laplacian is nonpositive and strictly negative on that support; sufficiently large $A$ makes both $q$ and $\varphi +q$ superharmonic. Their normalized Kelvin transforms are superharmonic even at interior infinity by the normalized Kelvin lemma. On the exterior ball the Kelvin multiplier is bounded by one (it is one in the plane), so uniform approximation survives. Each transformed approximant is a difference of bounded superharmonic traces on a compact neighborhood of $\overline\Omega $; bounded subharmonic trace resolutivity, proved by comparing its lower solution with the upper class, applies with the compactified minimum principle. Uniform stability of the Perron classes proves continuous-data resolutivity. Positive-functional representation gives a probability harmonic measure on the compact boundary, including when the pole is infinity.

For a polar boundary set, transform its finite points into the bounded ball and split it into countably many closed shells avoiding the center. The prescribed-point polar construction gives positive superharmonic pole functions finite at the center and at the chosen transformed interior point. Lower each harmonically on a sufficiently small center ball, which avoids that shell, and subtract its center value. Its normalized Kelvin transform is superharmonic at infinity and has the same required boundary blow-up. It is defined on a neighborhood of the compact exterior, and hence has a finite lower bound there by lower semicontinuity. Add a constant to make it positive. Choose summable positive coefficients giving a finite sum at the prescribed point (and at infinity if interior). Their sum is the required function. In dimension two, if the polar set contains infinity, add a positive multiple of $\log (|x-a|/R)$ on the exterior. In higher dimensions infinity cannot belong to a polar set. The same construction on a finite boundary singleton and epsilon comparison shows that its harmonic measure is zero.

A local finite-point weak barrier can be upgraded on the bounded set $D=\Omega \cap B(a,r)$ by the proved bounded-domain weak-barrier theorem. Its strong barrier $g$ has a positive lower bound away from $a$. On a cutting sphere inside $B(a,r)$ choose $c>0$ below that lower bound; paste $\min (g,c)$ inside to the constant $c$ outside. They agree on a collar, so the pasted function is superharmonic, vanishes at $a$, and is bounded away from zero off every neighborhood of $a$. If infinity is interior, its value $c$ dominates every exterior sphere mean, since it is globally at most $c$. Thus the pasting remains valid there. The usual upper/lower barrier comparison proves regularity. Conversely use a bounded continuous boundary function $m\ge 0$ vanishing only at $a$. Its harmonic solution is positive on every component: otherwise its probability harmonic measure would be concentrated on the zero-mass singleton. At a regular point it tends to zero, giving a weak barrier. This avoids unbounded distance data on a boundary containing infinity.

For a boundary infinity in $d\ge 3$, the fundamental function with any fixed finite pole is an explicit local weak barrier, positive and bounded away from zero on every bounded portion away from infinity. In dimension two inversion has multiplier one and transfers the finite-point local barrier criterion. At all finite points inversion preserves weak barriers locally. Thus the bounded-domain Kellogg theorem transfers to the inverted domain; polar sets transfer by Kelvin transformation. In $d\ge 3$ the possible extra center corresponds to an already regular infinity. This proves polar irregularity in all cases under the present nondensity hypothesis. $\square$