We say that a tempered distribution $f$ satisfies $f \in H^s(\mathbb R)$ for some $s \in \mathbb R$ if $(1+|\xi|^2)^{s/2} \hat f \in L^2(\mathbb R)$. Here, $\hat f$ denotes the Fourier transform of the tempered distribution.
My question: Can the Fourier transform be extended to a bounded mapping on the spaces $H^s(\mathbb R)$ into another Hilbert space, and what is the space it maps into? Can you provide a reference/proof?
Well, by definition the Fourier transform is an isometry between $H^s(\mathbb{R}^n)$ and $L^2(\mathbb{R}^n, (1+|\xi|^2)^sd \xi)$.