Newtonian potential. Let $f \in \mathcal{D}^{\prime}$. The convolutions
$$
V_{n}=\frac{1}{|x|^{n-1}} * f, \quad n \geq 3 ; \quad V_{2}=\ln \frac{1}{|x|} * f, \quad n=2
$$
(if they exist) are called the Newtonian (for $n=2$, the logarithmic) potential with density $f$.
The potential $V_{n}$ satisfies the Poisson equation
$$
\Delta V_{n}=-(n-2) \sigma_{n} f, \quad n \geq 3 ; \quad \Delta V_{2}=-2 \pi f .
$$
Indeed, using the formula of Sec. $2.4$ and, we obtain, for $n \geq 3$,
$$
\begin{aligned}
\Delta V_{n} &=\Delta\left(\frac{1}{|x|^{n-1}} * f\right)=\Delta \frac{1}{|x|^{n-2}} * f \
&=-(n-2) \sigma_{n} \delta * f=-(n-2) \sigma_{n} f
\end{aligned}
$$
We proceed in similar fashion in the case of $n=2$ as well.
MATH0070 COURSE NOTES ：
$$
\begin{aligned}
&a(x)=\sum_{|\alpha| \leq m} a_{\alpha} \partial^{\alpha} \delta(x) \
&a * u=\sum_{|\alpha| \leq m} a_{\alpha} \partial^{\alpha} u(x)
\end{aligned}
$$
linear difference equations:
$$
\begin{aligned}
a(x) &=\sum_{\alpha} a_{\alpha} \delta\left(x-x_{\alpha}\right), \
a * u &=\sum_{\alpha} a_{\alpha} u\left(x-x_{\alpha}\right)
\end{aligned}
$$