Timeline for Example for the Sobolev embedding theorem when p=n.
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Apr 26, 2012 at 19:18 | comment | added | Bazin | Well, may I say that you should get familiar with Fourier analysis. For $u\in \mathscr S(\mathbb R^n)$, you have $$ \hat u(\xi)=\int e^{-2i\pi x\cdot \xi} u(x) dx,\quad u(x)=\int e^{2i\pi x\cdot \xi} \hat u(\xi) d\xi, $$ and both formulas can be extended to temperate distributions, i.e. to the topological dual of $\mathscr S(\mathbb R^n)$. | |
| Apr 26, 2012 at 19:10 | comment | added | Giuseppe | Thanks. Do you know of a more explicit example or is this the only way? I am not familiar with Fourier inversion. | |
| Apr 26, 2012 at 16:08 | comment | added | Bazin | OK, but you can modify the definition of $\hat u$ above by multiplying the numerator by $ \mathbf 1(\xi_n\ge \vert\xi\vert) $ so that $\partial_n u$ will not be bounded. | |
| Apr 26, 2012 at 13:09 | comment | added | Willie Wong | It may not be entirely obvious that $\sqrt{(-\triangle)}u \notin L^\infty \implies \nabla u \notin L^\infty$ | |
| Apr 26, 2012 at 12:22 | history | answered | Bazin | CC BY-SA 3.0 |