Proposition 4.2.112BC493B5F349

Proof. We first prove the gradient identity for the partial sums of a decomposition of $\mu $ and $\nu $ into pieces with bounded Green potentials, to which proposition 4.1.29 applies, and then pass to the limit. Translating, we may assume that $B=B(0,R)$ for some $R>0$, and we choose $r<R$ such that the supports of $\mu $ and $\nu $ lie in $B(0,r)$. By lemma 4.1.14, the function $\kappa (x,y)=G_ B(x,y)-\Gamma _ d(x-y)$ extends to a continuous function on the compact set $\overline B\times \overline{B(0,r)}$; let $C$ be a bound for $|\kappa |$ there. Since $\Gamma _ d=-E_ d$, every finite positive measure $\lambda $ supported in $B(0,r)$ satisfies

and the last integral is a continuous function of $x\in \overline B$ bounded by $C\lambda (B)$.

Step 1: decomposition. Since $\int G_ B\mu \, \mathrm d\mu =I_ B(\mu )<\infty $, the Green potential $G_ B\mu $ is finite $\mu $-almost everywhere, and by the display so is $E_ d*\mu $. By part (2) of theorem 4.1.10, $\mu $ is therefore the sum of its restrictions to countably many disjoint compact sets, each of which has a finite and continuous fundamental potential. Let $\mu _ n$ be the sum of the first $n$ of these restrictions. Since $E_ d*\mu _ n$ is finite and continuous, the display shows that $G_ B\mu _ n$ is finite, continuous and bounded on $B$; this proves the last assertion of the proposition. By part (2) of proposition 4.1.16, $-G_ B\mu _ n$ is a subharmonic function on $B$ with Riesz measure $\mu _ n$, which extends harmonically across $\partial B$ and vanishes on $\partial B$; in particular, it tends to zero at $\partial B$. Thus proposition 4.1.29 applies to the functions $-G_ B\mu _ n$, $n\geq 1$. Hence $\nabla G_ B\mu _ n\in L^2(B)$ and

Step 2: the gradient of $G_ B\mu $. For $m>n$, expanding the square with the last display gives

where the inequality holds because $0\leq \mu _ m-\mu _ n\leq \mu -\mu _ n$ and $G_ B\geq 0$. The measure $\mu -\mu _ n$ is the restriction of $\mu $ to a decreasing sequence of sets whose intersection is $\mu $-null. Since $G_ B$ is nonnegative and integrable for $\mu \otimes \mu $, dominated convergence gives $I_ B(\mu -\mu _ n)\to 0$. Thus $(\nabla G_ B\mu _ n)$ is a Cauchy sequence in $L^2(B)$; let $g$ be its limit.

On the other hand, $G_ B\mu _ n\uparrow G_ B\mu $ pointwise by monotone convergence, and $G_ B\mu \in L^1(B)$. Indeed, for $y\in \operatorname{Supp}\mu $, the bound $|\kappa |\leq C$ and the local integrability of $\Gamma _ d$ (proposition 3.1.13) give

and integrating against $\mu $, by Tonelli's theorem, shows that $\int _ BG_ B\mu \, \mathrm d x$ is at most $\mu (B)$ times the right side. Hence $G_ B\mu _ n\to G_ B\mu $ in $L^1(B)$ by monotone convergence. For a smooth compactly supported vector field $\varphi $ on $B$ we obtain

so $g=\nabla G_ B\mu $. Since $\| \nabla G_ B\mu _ n\| _2^2=I_ B(\mu _ n)\uparrow I_ B(\mu )$ by monotone convergence, we get $\| \nabla G_ B\mu \| _2^2=I_ B(\mu )$.

Step 3: the mutual energy. Construct the partial sums $\nu _ n$ of $\nu $ in the same way. By proposition 4.1.29 and the Cauchy–Schwarz inequality in $L^2(B)$,

As $n\to \infty $, the left side increases to $I_ B(\mu ,\nu )$ by monotone convergence, so the mutual energy is finite, while the integral converges to $\int _ B\nabla G_ B\mu \mathbin {\cdot }\nabla G_ B\nu \, \mathrm d x$ by convergence in $L^2(B)$. This proves the identity for positive measures. For signed measures, we expand both sides bilinearly in the positive and negative parts.

Step 4: positive definiteness. Let $\sigma $ be a signed measure as in the statement, and put $G_ B\sigma =G_ B\sigma ^+-G_ B\sigma ^-$, a function in $L^1(B)$ by Step 2. Then $I_ B(\sigma )=\| \nabla G_ B\sigma \| _2^2\geq 0$. Suppose that $I_ B(\sigma )=0$, so that $\nabla G_ B\sigma =0$ as a distribution on $B$. Since $B$ is connected, corollary 2.5.5 shows that the distribution defined by $G_ B\sigma $ is a constant. On the other hand, $-\Delta G_ B\sigma =\sigma $ in the sense of distributions, by part (2) of proposition 4.1.16. Hence $\sigma ^+$ and $\sigma ^-$, which are Radon measures by proposition 2.2.2, define the same distribution, so $\sigma ^+=\sigma ^-$ by part (4) of proposition 2.5.3, that is, $\sigma =0$. Finally, the Cauchy–Schwarz inequality for $I_ B$ follows from the gradient identity and the Cauchy–Schwarz inequality in $L^2(B)$. $\square$