In the problem I am studying I have this quantity $$\| |\xi|^{s}\hat{f}\|_{L^{r'}(\mathbb{R}^d)}= \bigg( \int_{\mathbb{R}^d} \bigg | |\xi|^{s} \hat{f}(\xi)\bigg|^{r'} d \xi \bigg)^{\frac{1}{r'}}, \quad \textbf{r'<2}.$$
In Masaki, Satoshi; Segata, Jun-Ichi, On the well-posedness of the generalized Korteweg-de Vries equation in scale-critical lr-space, Anal. PDE 9, No. 3, 699-725 (2016). ZBL1342.35309.
authors define the space $$ \hat{H^r}_s(\mathbb{R}^d) :=\{ f \in \mathcal{S}'(\mathbb{R}^d), \ \|f \|_{\hat{H^r}_s(\mathbb{R}^d)}= \| (1+ |\xi|^2)^{\frac{s}{2}}\hat{f}\|_{L^{r'}(\mathbb{R}^d)}<\infty \}$$
So, If I take a function $f \in \hat{H^r}_s(\mathbb{R}^d)$ then $\| |\xi|^{s}\hat{f}\|_{L^{r'}} <\infty$ since $\| |\xi|^{s}\hat{f}\|_{L^{r'}}\leq \| (1+ |\xi|^2)^{\frac{s}{2}}\hat{f}\|_{L^{r'}(\mathbb{R}^d)}<\infty $
I would like to understand better which kind of function $f$ belongs to the space $\hat{H^r}_s(\mathbb{R}^d)$, it is very hard to me to think of an example. What are the sufficient assumptions to make on $f$ to get $f \in \hat{H^r}_s(\mathbb{R}^d)$? Can you suggest some books that deals with this "hat" Sobolev spaces?
The classical Bessel potential space is $$H^{r}_s(\mathbb{R}^d):=\{ g \in \mathcal{S}', \ \mathcal{F}^{-1}[(1+ |\xi|^2)^{\frac{s}{2}}\hat{g}] \in L^r(\mathbb{R}^d)\}$$ so it is reasonable to think that $\hat{H}^{r}_s$ is the Forier trasform space of the Bessel potential space $H^{r}_s$.
Let me start by saying I don't have an example yet. I'll edit if I can come up with one.
One thing you can say is a necessary condition for $f$ to belong to this hat space: By the Hausdorff-Young inequality we know $$ \| \check{h}\|_{L^r} \leq \| h\|_{L^{r'}}, \qquad 1\leq r'\leq 2. $$ Set $h=(1+|\xi|^2)^{s/2}\hat{f}(\xi)$ to see that $$ \| f\|_{H^r_s} \leq \| f\|_{\hat{H}^r_s}. $$ In particular $\hat{H}^r_s\subset H^r_s$ (with continuous embedding).
Another point of view:
We can characterize the Sobolev spaces $H^r_s$ in terms of Littlewood-Paley decompositions: If $\hat{\psi}_k(\xi)=\hat{\psi}(2^{-k}\xi)$ for $k\geq 1$ and $\hat{\psi}_0$ supported near the origin such that $\sum_{k\geq 0}\hat{\psi}_k(\xi) =1$ pointwise, we can define $\widehat{P_k f}(\xi)=\hat{\psi}_k(\xi) \cdot f(\xi)$, and then $$ \| f\|_{H^r_s} \approx \left\| \left(\sum_{k\geq 0} 2^{2ks}|P_k f(x)|^2\right)^{1/2}\right\|_{L^r}. $$ Now in your space we can decompose (recall that the support of $\hat{\psi}_k$ is in $|\xi|\approx 2^k$) $$ (1+|\xi|^2)^{s}|\hat{f}(\xi)|^2\approx \sum_{k\geq 0} 2^{2ks}|\widehat{P_kf}(\xi)|^2, $$ so that $$ \| f\|_{\hat{H}^r_s}\approx \left\| \left( \sum_{k\geq 0} 2^{2ks}|\widehat{P_kf}(\xi)|^2\right)^{1/2}\right\|_{L^{r'}}. $$ So I guess another way of looking at this is that $f\in \hat{H}^r_s$ ensures that these projections are "small enough" in both the $x$ and $\xi$ variables.