I'm a newbie and need reference for the following proofs:
1.Inverse of a Laplacian operator is compact in $L^2(\Omega)$ : Proof
2.If I've a general Hilbert-Schmidt Integral operator defined from one $L^p(\Omega_1)$ to another $L^q(\Omega_2)$ with all the exponents defined appropriately and the Kernel is too defined appropriately.
Detail: Chapter 8 and 8.15 of "H.W. Alt. Lineare Funktionalanalysis", Springer-Verlag, Berlin, $2002$. But this is in German and I'm not able to understand. So here we have to prove that the operator is compact as it's done in $L^2$ case. But here it's for a general $L^p$ space.