Let $M \subset \mathbb{R}^n$ be a $C^k$ ($k \geq 2$) compact hypersurface of dimension $n-1$ without boundary. Consider $X=M \times (0,\infty)$ which has boundary $\partial X = M \times \{0\}$.

I am looking for properties of Sobolev spaces on the space $X$. For example, I would expect a trace map $T:H^1(X) \to H^{\frac 12}(\partial X)$ to be continuous with a continuous right inverse, etc. Does these properties and things like Poincare inequality still hold?

More information about Sobolev spaces on such domains would be greatly appreciated. I asked at http://math.stackexchange.com/questions/1076687/sobolev-space-on-m-times-0-infty-m-compact-closed-manifold but posted it here as I didn't find any answer there. Thanks.

I did search for terms involving "semiinfinite" on Google but I couldn't find anything appropriate..