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## $f \in W^{s,2}$ then $\int_{\Bbb R^n} \xi_j^{2s} | \mathscr F f( \xi) |^2 d \xi < \infty$?

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 $(1+ | \xi |^2)^s (\mathscr F f)^2 \in L^1$.

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 Try out first what happens if you take a Schwartz function, en.wikipedia.org/wiki/Schwartz_space – András Bátkai Oct 16 at 18:30