while reading an article about Navier Stokes $L^2$ decay, I found a statement about the growth of a Fourier transform. In the simpler version, the statement is the following: $$ \text{if } f \in L^2(\mathbb{R}^3) \cap W^{-1,1}(\mathbb{R}^3) \; \Longrightarrow \;\vert \hat{f}(\xi) \vert \leq C \vert \xi \vert, $$ where $\hat{f}$ denotes the Fourier transform of $f$ (assume $f \colon \mathbb{R}^3 \to \mathbb{R}$). Do you know why it is true? (To check the article, see "$L^2$ decay for weak solutions of the Navier Stokes equations" by Maria Schonbek, fifth formula, page 7),
Maybe it is something very easy, but I am not an expert on negative Sobolev spaces. For me, the space $W^{-1,1}(\mathbb{R}^3)$ is already problematic. Indeed, in most of the literature (as "Sobolev Spaces" by Adams & Fournier) only the space $W^{-m,p'}(\mathbb{R}^3)= \left(W^{m,p}(\mathbb{R}^3)\right)^*$, with $\frac{1}{p} +\frac{1}{p'} =1,\; 1<p'\leq \infty$ is defined and characterized.
Any advice is greatly appreciated. Thank you in advance.