Using standard notation, we refer to $H^s(\mathbb R) = W^{s,2}(\mathbb R)$ to be the Sobolev Hilbert spaces. As is often the case, it's natural to then consider properties of functions in $H^s(\mathbb R)$ by looking at how it behaves in the Fourier domain, more specifically, we have that for any $s\in \mathbb{R}$,

$ \begin{align*} H^s(\mathbb{R}) = \{ g \in L_2(\mathbb{R}) : \int (1+t^2)^{s}|\mathcal{F} g(t)|^2dt <\infty\}, \end{align*} $

where $\mathcal{F}$ is the Fourier transform. In particular to note is that this works for $s<0$ as well, and so negative order Sobolev spaces are easily defined. In particular, one can define the norm of $H^s(\mathbb{R})$ to be defined by the relation $\|g\|_s = \int (1+t^2)^{s}|\mathcal{F}g(t)|^2dt$.

$\textbf{My first question:}$ I have seen, in particular the book Sobolev Spaces by Adams, Fournier (page 64), that another way to define the norm of the dual space of $H^s$, which appears to be $H^-s$ is given by the following: $$ \begin{align*} \|g\|^*_{-s} = \sup_{h\in H^{s}}\frac{\langle g,h\rangle}{\|h\|_s}, \end{align*} $$

where $\langle g,h\rangle$ is the standard $L_2$ inner product. I am having difficulty figuring out whether the norm defined by Fourier transformations or the one defined by using the dual are equivalent or not. The furthest I have been able to show is that $\|g\|^*_{-s} \leq (2\pi)^{-1}\|g\|_{-s}$ by Parceval's relation.

$\textbf{My second question:}$ If there appears to be a relationship, then I would like to restrict attention to spaces of the form $H^{s}(A)$ where $A\subset \mathbb{R}$ with defined norm $\|g\|_{s,A} = \inf\{ \|g^{'}\|_s : g^{'}_{|A} = g\}$ for all $g\in H^s(A)$. Then is there a relationship between $\|g\|_{s,A}$ and

$$ \begin{align*} \|g\|_{-s,A}^* = \sup_{h\in H^s(A)}\frac{\langle g,h\rangle_{A}}{\|h\|_{s,A}}, \end{align*} $$

where $\langle g,h\rangle_{A}$ is the standard $L_2$ inner product restricted to $A$? I am having difficulty trying to relate the two, because there seems to be a natural relationship globally, which I would hope to think there is a local relationship as well.

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