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.

  • $\begingroup$ I guess $W^{s,2}$ is defined by Fourier transform. In this case, first assuming that w is a test function, we can use inverse Fourier transform. $\endgroup$ – Davide Giraudo Sep 28 '12 at 9:09
  • $\begingroup$ Strictly speaking, you cannot reduce to the usual theorem, unless $s \in \mathbb{N}$. $\endgroup$ – Siminore Sep 28 '12 at 9:26

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 $2s>n$.


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