The Sobolev spaces are defined for $ 1\leq p < \infty$, as follows $$ W^{1,p} := \left\{ u\in L^p(\mathbb{R}) \,\Big\vert \exists \, g\in L^p(\mathbb{R}) \mbox{ such that } \int_{-\infty}^{\infty} u\varphi' = - \int_{-\infty}^{\infty} g\varphi ,\, \forall \varphi \in C^{1}_c(\mathbb{R}). \right\} $$
If we consider the Fourier transform \begin{equation} \mathcal{F}_p(f)(s):=\int_{\mathbb{R}} e^{-isx} f(x)\,dx \qquad (s\in \mathbb{R}) \end{equation} Then it is known that $\mathcal{F}_p(f)$ belongs to $L^q(\mathbb{R})$ for $1<p<2$ and $1/p + 1/q =1$.
I wonder know if there is something else we can say about the image of $W^{1,p}$ under the action of $\mathcal{F}_p$