An inquiry on Sobolev space

Question

Let $\widetilde{H}(\mathbb{R}^d) = \left \{ f \in L^2(\mathbb{R}^d): G(\xi)\mathcal{F}f \in L^2(\mathbb{R}^d) \right\}$ and $\widetilde{H}(\mathbb{R}^d)$ coincides with the Sobolev space $H^\beta(\mathbb{R}^d)$.

Is the following relationship established?

$m {(1+{\left \| \xi \right \|}_2^2)}^{\beta} \leq G(\xi) \leq M {(1+{\left \| \xi \right \|}_2^2)}^{\beta} \quad \left \| \xi \right \| > \xi_0$

for some $m,M >0$ and $\xi_0$.