Sobolev spaces on unbounded domains

Question

Sobolev function on unbounded domains. Let $H^s(\mathbb{R}^d),s\in\mathbb{R}$ be a fractional Sobolev space and let $f\in H^s(\mathbb{R}^d)$. Can we say that the restriction $f_{|A}$ of $f$ to $A$, some bounded subset of $\mathbb{R}^d$ belongs to $H^s(\mathbb{R}^d)$. This question came when I tried to understand whether functions on bounded sets can be considered as members of $H^s(\mathbb{R}^d)$. The difficulty that I have is that I don't know how the Fourier transform of restriction behaves.

Answer 1

If $\Omega \subset \mathbb{R}^d$ is an open set (possibly limited), you define $H_{0}^s(\Omega) = \overline{C_{c}^\infty(\Omega)}^{H^s}$ like closure with respect to $H^s$-norm, then it is clear that $H_{0}^s(\Omega) \subset H^s(\mathbb{R}^d)$. How the Fourier transform behaves is not so relevant. Hilbert-Sobolev spaces $H^s_{0}(\Omega)$ are defined in this way.