$L^2$ bound and Sobolev spaces

Question

Let $f \in L^2(\mathbb R)$ be a function such that

$$\vert f \vert_{\alpha}:=\sup_{h>0}h^{-\alpha}\Vert f(\bullet+h)-f \Vert_{L^2}< \infty$$

for some $\alpha \in (0,1).$

I would like to know whether there exists $\beta \in (0,1)$ such that $\vert f \vert_{\alpha}$ satisfies for some constant $C_{\alpha,\beta}>0$

$$ \vert f \vert_{\alpha} \le C_{\alpha,\beta} \left\lVert \langle \bullet \rangle^{\beta} \widehat{f} \right\Vert_{L^2}$$

for all $f$ for which the right-hand side is finite.

where $\langle x \rangle:=\sqrt{1+\vert x \vert^2}.$

Answer 1

This works for $\beta\ge\alpha$. In terms of the Fourier transform $g=\widehat{f}$, what you're trying to establish becomes $$ h^{-2\alpha}\int |g(t)|^2 \left| e^{ith}-1\right|^2 \, dt \lesssim \int |g(t)|^2 (1+t^2)^{\beta}\, dt . $$ We can see that this holds by considering separately $|t|\ge 1/h$ and $|t|<1/h$ in the first integral. In the first contribution, estimate $|e^{ith}-1|\le 2$, and in the second one, use $h^{-\alpha}|e^{ith}-1|\lesssim |t|h^{1-\alpha}\le |t|^{\alpha}$.