Sobolev spaces on Riemannian manifolds - Reference request

Question

I need to study Sobolev spaces on manifold $M=\mathbb{R}^n$ endowed with the riemannian metric $g$. Is there a book or paper that talks about Sobolev spaces ( $W^{m,p}(M,g)$ or $H^{m,p}(M,g)$ ) on such a manifold? I am interested more in the case $p=2$. Texts that I came across deal with only compact manifolds.

Any help is deeply appreciated.

Answer 1

Please look at the book

E. Hebey (1999) Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities

In Chapter 2 he is dealing with the compact case and in Chapter 3 with the non-compact one. I think this is exactly what you want. The Sobolev space $H^{k,p}(M,g)$ is defined for a non-compact Riemannian manifold $M$ and all kinds of useful results are derived.