3.5.7 Potentials and global Riesz decomposition47255929FFC0

Proof. Fatou's lemma applied to the nonnegative continuous extended kernel gives lsc; Tonelli and the supermean inequality for each pole give supermeans. If $\mu $ is finite and gives zero mass to a closed interior ball containing $x$, the bound for $G(x,\cdot )$ outside that ball makes its potential finite at $x$, hence superharmonic on that component. Split a finite measure into its part on a small interior ball and its complement. Each part misses some interior ball in the component, so both have superharmonic potentials and their sum is superharmonic. Apply this separately to every component, using that $G$ vanishes between components.

A potential $p$ is locally integrable by superharmonic local integrability. For $\varphi \in C_ c^\infty (\Omega )$,

Thus signed Fubini is legitimate, and the fundamental-solution identity plus harmonicity of the correction gives $\int p\Delta \varphi =-c_ d\int \varphi \, d\mu $. In particular a harmonic potential has zero measure; more generally subtracting the distributional identities proves equality of measures wherever the difference is harmonic. If $\mu (U)=0$, Weyl's lemma gives a harmonic representative on $U$; canonical representative uniqueness identifies it pointwise with $p$.

For the greatest minorant use the same sequence of harmonic lowerings for $p$ and for every kernel $G(\cdot ,y)$. Induction and nonnegative Tonelli show $p_ j(x)=\int G_ j(x,y)\, d\mu (y)$. For each $y$, $G_ j(x,y)\downarrow 0$, because the greatest harmonic minorant of the minimal Green kernel is zero. In each connected component choose $x_0$ with $p(x_0)<\infty $. Dominated convergence gives $p_\infty (x_0)=0$; its nonnegative harmonic minorant is therefore zero on that component by the minimum principle. This is a componentwise argument, even when $\Omega $ is disconnected.

On $S$, domain monotonicity gives $0\le G_ S\mu \le p$. Both potentials are locally integrable; their distributional difference is harmonic and nonnegative almost everywhere, hence a nonnegative harmonic $h$. Equality $p=G_ S\mu +h$ holds everywhere by uniqueness of superharmonic representatives. Finally symmetry of $G$ and nonnegative Tonelli give reciprocity, including infinite values. None of these arguments separates divergent fundamental and harmonic-correction integrals. $\square$

Proof. The checked distributional characterization gives the positive Radon measure $\nu $. We first suppose $u$ is bounded below by a constant $m$ on its Greenian domain $D$ (either $D=W\Subset \Omega $, or $D=\Omega $ and $m=0$). For a compact $K\subset D$ put $p_ K=G_ D(\nu |_ K)$, a potential of a finite measure. The locally integrable difference $u-p_ K$ has nonpositive Laplacian, so it has a superharmonic representative $w_ K$, and $u=w_ K+p_ K$ everywhere by canonical representative uniqueness.

The function $p_ K$ is bounded outside some compact neighborhood $K'\Subset D$. Here is the required uniformity: cover $K$ by finitely many small balls whose closures lie in $D$. On each, Harnack chains bound $G_ D(x,y)$ for $y$ in its smaller concentric ball by a fixed multiple of $G_ D(x,y_ i)$, whenever $x$ lies outside the union of the larger balls. Each fixed-pole kernel is bounded away from its pole; integrating the resulting finite uniform bound against $\nu |_ K$ proves the assertion. Choose $K'$ containing these larger closed balls. On $D\setminus K'$, $w_ K=u-p_ K\ge m-M$; on $K'$ it is bounded below by lsc and compactness. Hence $w_ K$ has a constant harmonic minorant on all of $D$.

The greatest-minorant operation is additive, and $p_ K$ has greatest minorant zero. Thus $H_ Du=H_ Dw_ K\ge m$ because $u\ge m$. Consequently $w_ K\ge m$ and $p_ K\le u-m$. Exhaust $D$ by increasing compact $K_ j$. Monotone convergence gives $p=G_ D(\nu |_ D)\le u-m$. This is finite almost everywhere, hence a potential. Its Riesz measure is $\nu |_ D$, so $u-p$ is harmonic in distributions and has a harmonic representative $h$. Canonical uniqueness gives $u=p+h$ everywhere, and additivity of greatest minorants makes $h=H_ Du$. When $m=0$, $h\ge 0$. Applying this construction to every relatively compact $W$ proves the local assertion without assuming $u$ bounded below on the entire $\Omega $. Uniqueness of $\nu $ follows from its distributional identity, and the last characterization follows from the zero-minorant property of potentials. $\square$