Theorem 4.5.130E65F552B1A6

Proof. Compactness of $M$ and the interior strong minimum principle prove the boundary minimum principle, so the upper/lower Perron constructions and harmonic alternatives apply. If $y\in \Delta _1$ and $U$ is a neighborhood of $y$ in $M$, then $p=\widehat R_{\Omega \setminus U}K_ y$ is a potential. Indeed its greatest harmonic minorant, being below the minimal $K_ y$, is $tK_ y$, $0\le t\le 1$. The case $t=1$ would represent the extreme $K_ y$ on $M\setminus U$, impossible by the support assertion. Write $p=w+tK_ y$ with $w$ a potential. Idempotence and additivity of reduction give $w+tK_ y=\widehat R_{\Omega \setminus U}w+tw+t^2K_ y$. Uniqueness of Riesz decomposition yields $t=t^2$, hence $t=0$.

Extend $f$ continuously to $M$ and first assume $0\le f\le 1$. Partition $\Delta _1$ into the bins $E_ i=\{ (i-1/2)/j\le f<(i+1/2)/j\} $, $0\le i\le j$, and put $h_ i=K(\chi |_{E_ i})$, so $\sum h_ i=1$. Let $F_ i=\{ x\in \Omega :|f(x)-i/j|\ge 1/j\} $. The preceding local-potential assertion and the integral-sweeping lemma make $p_ i=\widehat R_{F_ i}h_ i$ a potential. On $F_ i$, $p_ i=h_ i$ quasi-everywhere; off $F_ i$, $(i-1)/j<f<(i+1)/j$. Therefore

Initially the inequalities hold quasi-everywhere; canonical ball averages and continuity of $f$ give them everywhere. The left and right sides are bounded lower and upper competitors respectively. Taking harmonic minorants removes their potential terms: every harmonic minorant of a potential is at most zero. Thus the lower/upper solutions are between $\sum _{i\ge 1}(i-1)h_ i/j$ and $\sum _{i\ge 0}(i+1)h_ i/j$, a gap at most $2/j$. The same bounds enclose $K(f\chi )$. Letting $j\to \infty $ proves the formula, and scaling and adding constants handle arbitrary continuous data.

For the ball, expand $G(r\theta ,x)$ in $R-r$, uniformly for $\theta $ near a specified boundary point. The reference normal derivative is nonzero and bounded away from zero there; the quotient therefore tends along every approach to the quotient of normal derivatives. The explicit ball kernel gives precisely the normalized Poisson kernel displayed above. It separates sphere points, so uniqueness of function compactification identifies $M$ with the closed ball. We also verify the uniqueness needed in this example. For continuous sphere data $g$, the symmetry $P_ B(t\zeta ,y)=P_ B(ty,\zeta )$ and the ball approximate-identity estimate give

uniformly in $y$ as $t\uparrow 1$. Uniformity follows from uniform continuity of $g$, kernel mass one and a uniformly vanishing distant tail. Finite signed Fubini therefore recovers every continuous integral of a boundary measure from its interior Poisson function; equal functions have equal boundary measures. Now if $0\le h\le K_ y$, represent both $h$ and $K_ y-h$ by positive ball boundary measures. Their sum is the point mass representing $K_ y$, by this uniqueness, so each is supported on $\{ y\} $. Thus the sphere kernels are minimal and $\Delta _1=\Delta $. This argument supplies uniform control even for tangential approaches. $\square$