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Ignoring fractional sobolev spaces, if we restrict ourselves to $k>0$ when $k$ is an integer, then the Sobolev space of order $k$, for $W^{k,p}(\mathbb{R})$ is the space of functions $f$ such that $\|f\|_{W^{k,p}(\Omega)} \asymp \|f\|_{p} + \|f^{(k)}\|_{p}$ is finite.

A standard way of defining Sobolev spaces when $k<0$ is to say that if $1/p + 1/p^\prime = 1$, then $W^{-k,p^\prime}(\mathbb{R})$ is the dual space of $W^{k,p}(\Omega)$. In particular, when $p = 2$, we get a Hilbert space. For the rest of this question, we will assume $p=2$

My $\textbf{Question}$ is that there seems to be a lot of literature that considers defining a Hilbert Scale to be a sequence of embedded spaces, $H_{k+1} \subset H_k$, such that $H_k = \{f : \int (1+t^2)^k|\hat f(t)|^2dt <\infty \}$ where $\hat f$ is the Fourier transform of $f$. Naturally, when $k>0$ in this case, $H_k$ matches up with the Sobolev spaces $W^{k,p}(\mathbb{R})$, but I am not sure when $k<0$.

When $k<0$, do the definitions of the spaces $H_k$ and $W^{-k,2}(\mathbb{R})$ coincide?

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Yes, the definitions are the same. Consider the weighted Lebesgue space $L^2_k$ with the norm given by $$\|g\|_{k,2}^2 = \int_{\mathbb R} (1+t^2)^k|g(t)|^2\,dt $$ The dual of $L^2_k$ is naturally identified with $L^2_{-k}$, via the pairing $$ \langle g,h\rangle = \int gh = \int \left[(1+t^2)^{k/2}g(t)\right] \left[(1+t^2)^{-k/2}h(t)\right] $$ (Another way to put this: the map $L^2_k\to L^2 $ defined by $g\mapsto (1+t^2)^{k/2}g$ is an isometric isomorphism. Its adjoint is also of the form $g\mapsto (1+t^2)^{k/2}g$, but it maps $L^2$ onto $L_{-k}^2$.)

The Fourier transform is an isometric isomorphism between $H^k$ and $L^2_k$. Therefore, the dual of $H^k$ can be identified with $H^{-k}$.

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