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 $C_{\alpha,\beta}>0$

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

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

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

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