In our lecture we have shown that $\forall f \in L^2(\mathbb{R}^n_+) $ there is a unique $ u $ in the Sobolev space $ H^2(\mathbb{R}^n_+) $ satisfying $ -\Delta u = f. $
Now in our exercise sheet we are asked to show that there is an integral kernel $ \Phi $ such that $ u(x) = \int_{\mathbb{R}^n} \ \Phi (x-y) \ f(y) \ dy $.
Wikipedia tells me that there is an integral kernel and that it is of the form \begin{equation*} \Phi(x) \ = \ const. \ \cdot \ \frac{x_n}{({\sum_{i = 1}^{n} x_i^2})^{n/2}} \end{equation*}
So now to my question:
How can you show that this is indeed an integral kernel for poisson's equation? In particular, how can you differentiate under the integral sign and "take the Laplacian" of $ \Phi $ at $ x - y = 0 $ ? Moreover, do you know a priori that there has to be such an integral kernel?
Thanks a lot in advance, I would really appreciate your help!
Best regards
Phil