By Plancherel's Theorem, we can define the Sobolev space $W^{k,2}(\mathbb{R})$ using Fourier transforms as
$$W^{k,2} (\mathbb{R}) = \left\{ f \in \mathcal{S}' \ | \ \mathscr{F}^{-1}\left[ \left( 1 + \lvert \xi \rvert^2 \right)^{k/2} \mathscr{F}[f]\right] \in L^2(\mathbb{R}) \right\}$$
equivalently to the definition for general $p\ge 1$
$$W^{k,p}(\mathbb{R}) = \left\{ f \in L^p(\mathbb{R}) \ | \ \frac{d^if}{dx^i} \in L^p(\mathbb{R}), \ 1 \le i \le k \right\} \, ,$$
if we set $p=2$.
My question is whether it is possible to define $W^{k,p}(\mathbb{R})$ using Fourier transforms when $p \ne 2$. Using the Riesz-Thorin interpolation theorem as discussed in this MathSE question and answer, we can extend the Fourier transform to a (non-surjective) bounded linear transformation $L^p \to L^q$ for $1 \le p \le 2$ (where $q$ is the Holder conjugate of $p$). As discussed in this book chapter (p178), we can invert the Fourier transform on its range in $L^q$.
Is this enough that we can define Sobolev spaces for $1 \le p < 2$ in terms of Fourier transforms as we can when $p=2$?
