Show $u$ satisfy poisson equation Let $f\in \mathcal S(\mathbb R^n),n\geq 3$. How can I show that $$u(x)=C_n\int_{\mathbb R^n}|x-y|^{2-n}f(y)dy$$
where $C_n$ is a suitable constant solve Poisson equation ? i.e. $$\Delta u=f.$$

The big problem is to permute $\frac{\partial }{\partial x_i}$ and the integral, but also if I can, I still have the fact that $|x-y|^{2-n}$ is not differentiable (and also not defined) when $y=x$.
This is proved in Evans's PDE book, where the necessity of $n&gt;2$ is also well explained.

Proof (Evans; I adapt.): The right-hand side of your formula provides a solution to $$\Delta u=f .$$ Because $f$ is Schwartz and

$$\int|x-y|^{2-n} \longrightarrow 0$$

outside large enough balls (note that this holds if $n&gt;2$), we can see that the right hand side is in fact a bounded solution to $\Delta u=f$.

By uniqueness of bounded solutions (Liouville's Theorem) you can fix the constant so that $u$ is of the right hand side form.

About how you prove that RHS "is" indeed a solution, you have to prove that despite the discontinuity at $x=y$ the integral is continuously differentiable twice, and then calculate the Laplacian. You will then prove that it equals f. Fix an $x$, write the very definition of partial derivative as $h\longrightarrow 0$ of the difference quotient. In all other calculations, again work with fixed $x$.

Hope this helps. I know I did not present in the most clear way, but... Feel free to ask.
This is proved in Evans's PDE book, where the necessity of $n&gt;2$ is also well explained.

Proof (Evans; I adapt.): The right-hand side of your formula provides a solution to $$\Delta u=f .$$ Because $f$ is Schwartz and

$$\int|x-y|^{2-n} \longrightarrow 0$$

outside large enough balls (note that this holds if $n&gt;2$), we can see that the right hand side is in fact a bounded solution to $\Delta u=f$.

By uniqueness of bounded solutions (Liouville's Theorem) you can fix the constant so that $u$ is of the right hand side form.

About how you prove that RHS "is" indeed a solution, you have to prove that despite the discontinuity at $x=y$ the integral is continuously differentiable twice, and then calculate the Laplacian. You will then prove that it equals f. Fix an $x$, write the very definition of partial derivative as $h\longrightarrow 0$ of the difference quotient. In all other calculations, again work with fixed $x$.

Hope this helps. I know I did not present in the most clear way, but... Feel free to ask.