Theorem 3.5.99453A8C660CC

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. $\square$