4.4.14 Perron theory on the one-point compactification02718F5DB5E5

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. $\square$

Proof. Only the dense case remains. Work in a connected component, choose $a\in \Omega $, and concentric closed balls with $0<r_0<r_1<r_2<r_3$, all contained in $\Omega $. Removing the smallest ball gives a nondense domain. Its pole function $v$ can be harmonically lowered on the annulus $r_0<|x-a|<r_3$, so it is finite harmonic near the two middle spheres. This preserves its true-boundary blow-up and positivity. Write $O=\Omega \setminus \overline{B(a,r_1)}$. This set is connected: any path in $\Omega $ entering the removed ball can be rerouted through the contained annulus. The harmonic measure $Q$ of its inner sphere is strictly below one on $O$. Indeed the original nonpolar boundary contains a regular point, since the nondense Kellogg exception is polar; $Q$ tends to zero there. The strong maximum principle then gives $q=\max _{|x-a|=r_2}Q(x)<1$.

Set $w_0=0$. Given $w_{j-1}$, let $u_ j$ be the ball solution in $B(a,r_2)$ with data $v+w_{j-1}$ on its sphere. Let $w_ j$ be the bounded solution on $O$ with data $u_ j-v$ on the inner sphere and zero on the true boundary. These are continuous boundary data on the nondense domain. For consecutive differences the ball maximum principle and the bound by $Q$ give

Thus the corrections converge in geometric series, uniformly on the respective boundaries and hence throughout their domains. Their limits $u,w$ are harmonic; $w$ is bounded. On both annulus boundary spheres, $u=v+w$, so annular uniqueness gives this equality throughout the overlap. Paste $u$ inside to $v+w$ outside. The result is superharmonic, harmonic near $a$, bounded below, and has the same polar boundary blow-up as $v$. Adding a constant makes it positive. This proves the assertion including finiteness at the prescribed center. Componentwise construction proves the disconnected version.

For bounded harmonic $h$ with zero boundary limits outside a polar $Z$, compare $\pm h+\varepsilon p$ with zero, where $p$ is the positive blow-up function, finite at a prescribed interior point. Compactified boundary comparison applies even on $Z$, and yields $|h(a)|\le \varepsilon p(a)$. Let $\varepsilon \downarrow 0$ and choose $a$ arbitrarily. The same epsilon argument permits polar exceptions in bounded Perron competitors. $\square$

Proof. For a dense domain remove an interior closed ball. Solve the continuous datum $f$ on the true boundary and the constant $M+1$ on the artificial sphere, where $|f|\le M$. Paste $M+1$ inside by taking the minimum with that constant on a collar. The resulting bounded superharmonic function is a global upper competitor. The lower construction uses $-M-1$. At every regular true boundary point of the punctured nondense domain, these give the corresponding boundary limits for the global upper/lower solutions. Their difference is bounded harmonic, with zero boundary limits outside the polar Kellogg set, and is zero by the preceding lemma. Thus $f$ is resolutive. Those same upper/lower bounds show each such punctured-domain regular point is regular for $\Omega $, proving Kellogg for the general domain. Local barriers and the capped-constant construction give the equivalent local criterion as before.

The full measurable-data proof now uses precisely the steps established for bounded domains: closed boundary indicators by continuous decreasing approximation, the pi-lambda argument, nonnegative simple approximation, and monotone continuity of the upper envelope via summable positive errors at a fixed pole. The only local harmonic operations needed are lowering and monotone convergence, available for both interior and exterior balls. For clarity, the necessity step also survives unchanged: countably many upper/lower competitors approximating a finite solution at a pole give Borel boundary liminf/limsup sandwiches. Their integrals have the same value at that pole, so their difference is zero almost everywhere there; positive harmonicity and Harnack transfer this common null set throughout the component. The sandwiches identify the datum in the completed measure class and give its absolute integrability. Conversely positive and negative integrable parts follow from monotone simple functions, and null changes are removed by the sandwich comparison. Apply this separately in the countably many components. Epsilon comparison with the constructed pole functions proves polar boundary nullity. This establishes the same full equivalence as the bounded Perron theorem, now with the compactified boundary condition at infinity included explicitly. $\square$

Proof. The integral solution is bounded above by $u$, which is an upper competitor, and so is finite almost everywhere in each component. Positive harmonic alternatives make it finite everywhere there. Paste $H_{u_0}$ off $F$ to $u$ on $F$ and regularize. The ball comparison argument of the relatively compact sweeping theorem applies: restriction consistency compares on the intersection of each interior ball with $\Omega \setminus F$, and regular boundary limits follow by increasing continuous minorants of $u_0$. The exceptional set is precisely among the irregular interior boundary points, independently of $u$. The regularized paste is therefore superharmonic and majorizes $u$ on $F$ quasi-everywhere. Every defining majorant for $R_ Fu$ is an upper Perron competitor off $F$; the two inequalities identify the reduction. Linearity of nonnegative boundary integrals gives additivity off $F$; on $F$ both sides equal the original sum quasi-everywhere. Canonical ball averages remove this polar exception from the identity everywhere. $\square$