Proposition 3.4.111F52C5AF98EF

Proof. By proposition 2.2.2, $\nu $ is a Radon measure; let $L$ be its compact support, so that $\nu (\mathbb R^ d\setminus L)=0$ (section 2.2). For each $x$, the function $E_ d(x-\cdot )$ is bounded above on $L$, so $p(x)$ is well defined in $[-\infty ,\infty )$. Next, $E_ d$ is subharmonic on $\mathbb R^ d$. For $d\geq 2$ it is harmonic on $\mathbb R^ d\setminus \{ 0\} $; it is upper semicontinuous at the origin because $E_ d(x)\to -\infty $ as $x\to 0$, and the submean inequality at the origin holds because $E_ d(0)=-\infty $. Since subharmonicity is local, $E_ d\in \mathrm{SH}(\mathbb R^ d)$. For $d=1$, the triangle inequality gives $|x|\leq \frac12(|x+r|+|x-r|)$, which is the submean inequality for $E_1=|x|/2$.

To show that $p$ is subharmonic, we truncate the kernel. The functions $E_ d^{(m)}=\max \{ E_ d,-m\} $, $m\in \mathbb N$, are continuous and subharmonic, and they decrease to $E_ d$. Put $p_ m=E_ d^{(m)}*\nu $. For $x$ in a compact set $K$, the differences $x-y$ with $y\in L$ lie in the compact set $K-L$, on which $E_ d^{(m)}$ is uniformly continuous; hence $p_ m$ is continuous. By theorem 3.4.5, each translate $E_ d^{(m)}(\cdot -y)$ satisfies the submean inequality on every sphere. Integrating in $y$ and using Fubini's theorem for the bounded continuous integrand gives $p_ m(x)\leq M_{p_ m}(x,r)$ for all $x$ and $r>0$, so $p_ m\in \mathrm{SH}(\mathbb R^ d)$. For fixed $x$, the functions $E_ d^{(m)}(x-y)$ decrease to $E_ d(x-y)$ and are bounded above, uniformly in $m$ and $y\in L$, by the finite constant $\max \{ 0,\sup _{y\in L}E_ d(x-y)\} $. Monotone convergence, applied to the differences from this constant, gives $p_ m(x)\downarrow p(x)$. Hence $p$ is subharmonic by theorem 3.4.9.

To prove local integrability, let $K$ be compact and choose $R>0$ with $K-L\subset B(0,R)$. Since $E_ d$ is locally integrable, Tonelli's theorem gives

Hence $p$ is locally integrable. Finally, let $\chi $ be a test function. The preceding estimate with $K=\operatorname{Supp}\chi $ justifies the use of Fubini's theorem, and the identity $\Delta E_ d=\delta _0$, translated to the point $y$, gives

This is the distributional equation $\Delta p=\nu $. $\square$