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I want to know that how the statement below holds.

The statement : There exists a constant $C = C(s)$ such that the continuous embedding of $W^{s,2}$ into the space of uniformly bounded, continuous functions if $s > n/2$, i.e., $ |w(x)| \leqslant C \| w\|_{s,2}$ for $w \in W^{s,2}$ and almost all $x \in \mathbb R^n$.

Would you tell me how this holds by using the usual Sobolev embedding theorem? Thank you.

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    Strictly speaking, you cannot reduce to the usual theorem, unless $s \in \mathbb{N}$.2012-09-28

1 Answers 1

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We only need to prove $\hat{u}\in L^1$, then use the Fourier inverse transform.

Try this:

$(\int_{-\infty}^{\infty}|\hat{u}|dy)^2\leq \int_{-\infty}^{\infty}\frac{dy}{|y|^{2s}}\int_{-\infty}^{\infty}|y|^{2s}|\hat{u}|^2 dy\leq C||u||_{H^s}^2$

since $2s>n$.