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What is a sufficient condition for $s>0$ such that

$$\mathcal{F} \dot{H^s} (\mathbb{R}^n) \hookrightarrow L^2(\mathbb{R}^n)$$

where $\dot{H^s} (\mathbb{R}^n)$ is the homogeneous sobolev space: $\dot{H^s}= (- \Delta)^{\frac{s}{2}} L^2$,

$\mathcal{F}$ stand for the Fourier transform,

$\mathcal{F} \dot{H^s} =\{f \in \mathcal{S}^{\prime} ; \mathcal{F}^{-1} f \in \dot{H^s} \}$, and

$||f||_{\mathcal{F} \dot{H^s} } = ||\ |x|^s f\ ||_{L^2}$.

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    It seems this is mainly a problem of notation. You meant $||f||_{\mathcal{F} \dot{H^s} } = ||(1+ |x|^s) f\ ||_{L^2}$ ? – 2017-02-17
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    No, $||f||_{\mathcal{F} \dot{H^s} } = ||\ |x|^s f\ ||_{L^2}$. – 2017-02-17
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    In that case there is no inclusion. – 2017-02-17

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