If $f \in W^{s,2} (\Bbb R^n) $, then by the Plancherel's theorem, I know that its Fourier transform $ \mathscr F f(\xi) \in L^2 (\Bbb R^n) $. ($ \scr F$ means the Fourier transform). Now I want to show that if $ f \in W^{s,2} (\Bbb R^n)$, for $ \xi = (\xi_1 , \cdots ,\xi_n ), \; \text{integer}\; s \geqslant 0$, $$ \int_{\Bbb R^n} \xi_j^{2s} | \mathscr F f( \xi) |^2 d \xi < \infty \;\;(\forall j = 1,\cdots,n). $$ How can I prove this, or do I need some more assumptions? ($W^{s,p}$ means general Sobolev space)
If this holds then I think I can conclude that $(1+ | \xi |^2)^s (\mathscr F f)^2 \in L^1$.
