Lemma 4.1.2559F83B6EAAB7

Proof. Choose a cutoff $\chi \in C_ c^\infty (\Omega )$ with $0\leq \chi \leq 1$ and $\chi =1$ on an open neighborhood $V$ of $\operatorname{Supp}\sigma $, and put $\nu _ j=\chi \Delta v_ j$ and $\nu =\chi \Delta v$. Since $\Delta v_ j\to \Delta v$ distributionally, proposition 2.6.1 shows that $\nu _ j\to \nu $ vaguely, and the total masses $\nu _ j(\Omega )=\int \chi \, \mathrm d\Delta v_ j$ are bounded. By the Riesz decomposition (theorem 3.4.14), $v_ j=E_ d*\nu _ j+h_ j$ and $v=E_ d*\nu +h$ on $V$, with $h_ j$ and $h$ harmonic.

We show that $h_ j\to h$ uniformly on compact subsets of $V$. For a test function $\varphi $, Fubini's theorem gives $\int (E_ d*\nu _ j)\varphi \, \mathrm d x=\int (E_ d*\varphi )\, \mathrm d\nu _ j$, and $E_ d*\varphi $ is continuous by proposition 3.1.13; since the $\nu _ j$ are supported in the fixed compact set $\operatorname{Supp}\chi $, it follows that $E_ d*\nu _ j\to E_ d*\nu $ distributionally, and hence $h_ j\to h$ distributionally on $V$. The functions $h_ j$ are also bounded in $L^1$ on compact subsets of $V$. This holds for the $v_ j$, which converge in $L^1_{\mathrm{loc}}$, and for the $E_ d*\nu _ j$: if $Q\subset V$ is compact and $s>0$ is such that $Q-\operatorname{Supp}\chi \subset B(0,s)$, then Tonelli's theorem gives

and the masses $\nu _ j(\Omega )$ are bounded.

By the interior estimates (3.1.8), the derivatives of the $h_ j$ are therefore bounded on compact subsets of $V$, so the $h_ j$ are locally equicontinuous. Moreover, $h_ j\to h$ pointwise on $V$. Indeed, if $x\in V$ and $\varepsilon >0$ is less than the distance from $x$ to $\mathbb R^ d\setminus V$, then $h_ j(x)=(h_ j*\rho _\varepsilon )(x)$ and $h(x)=(h*\rho _\varepsilon )(x)$ by the last assertion of theorem 3.1.7, and $(h_ j*\rho _\varepsilon )(x)\to (h*\rho _\varepsilon )(x)$ by distributional convergence. To upgrade this to uniform convergence, let $Q\subset V$ be compact and $\theta >0$. By equicontinuity, every point of $Q$ has a neighborhood on which each $h_ j$, and also the continuous function $h$, varies by less than $\theta $. Finitely many of these neighborhoods, centered at points $x_1,\ldots ,x_ N$, cover $Q$, and for all large $j$ we have $|h_ j(x_ i)-h(x_ i)|<\theta $ for every $i$; hence $|h_ j-h|<3\theta $ on $Q$.

Next, the reciprocity law (4.1.2), applied to $\sigma $ and $\nu _ j$ and to $\sigma $ and $\nu $, gives

where the convergence holds because $E_ d*\sigma $ is continuous and the $\nu _ j$ converge vaguely with supports in $\operatorname{Supp}\chi $. All these integrals are finite, since $E_ d*\sigma $ is bounded on $\operatorname{Supp}\chi $. Adding the integrals of $h_ j$, which converge to $\int h\, \mathrm d\sigma $, proves that $\int v_ j\, \mathrm d\sigma \to \int v\, \mathrm d\sigma >-\infty $.

By Hartogs' lemma (theorem 3.4.20), $\varlimsup _ jv_ j\leq v$ everywhere, and the $v_ j$ are bounded above on $\operatorname{Supp}\sigma $ by a common constant for large $j$. Reverse Fatou's lemma therefore gives

Since $v-\varlimsup _ jv_ j\geq 0$ and $\int v\, \mathrm d\sigma $ is finite, it follows that $\varlimsup _ jv_ j=v$ $\sigma $-almost everywhere. $\square$