Let $f\in C^{\infty}(\mathbb{R}^N)$ be a function that aniihilates itself (is null) in the complementar of a compact of $\mathbb{R}^N$. Show that $$u(x) = \int_{\mathbb{R}^N}T_y(x)f(y)dy, \ x\in\mathbb{R}^N$$ is of class $C^{\infty}$ and that $\Delta u = f$ in $\mathbb{R}^N$
In the context of the Representation Theorem, we find a way to represent a function $u$ such that $\Delta u = f$ inside the domain and equal $f$ in the border. I've seen ways on how to arrive at the function that when integrated together with $f$ gives $u$. It was related to functions $\frac{1}{|x-y|^{N-2}}$. These functions don't annihilate outside a compact, but solve the problem.
Thinking a little bit, I must show
$$\Delta u(x) = \int_{\mathbb{R}^N}\Delta T_y(x) f(y) \ dy = f(x)$$
I know from proofs about the Representation Theorem that if $T$ is such that $\Delta T$ is the dirac delta function, we have the result. But I don't see why it works in this case and this is neither a proof, it must be an analytical proof.
UPDATE:
I just noticed it doesn't say anything about what is $T_y$, so I guess it's the green function without the corrector function. Maybe?
UPDATE:
$$T_y(x) =- \frac{1}{2\pi}\log |x-y|, \mbox{ for } N= 2\\ \frac{1}{\omega_N}\frac{1}{|x-y|^{N-2}}, \mbox{ for } N\ge 3$$