Sobolev spaces and compactly supported functions.

Question

It is well-known that $C^{\infty}_{c}(\mathbb{R}^{d})$ are dense in the Sobolev spaces $H^{k}(\mathbb{R}^{d})$. If I consider more generally the spaces $H^{s}(\mathbb{R}^{d})$ for $s\in\mathbb{R}$, is it then still true? For me its, not even clear that $C^{\infty}_{c}(\mathbb{R}^{d})\subset H^{s}(\mathbb{R}^{d})$ in the general case. Does anyone have a reference for this?

Answer 1

For $s>0$, the whole space the inclusion and the density are both true. You can find a proof in Adams' book "Sobolev Spaces", Theorem 7.38.