Poincaré inequality for Fractionnal Sobolev space

Asked Oct 25 '17 at 12:27

Active Oct 25 '17 at 12:27

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Is there a sort of Poincaré's inequality for Fractional Sobolev space ?

Something as $$\left\|u-\frac{1}{|\Omega |}\int_\Omega u\right\|_{W^{s,p}(\Omega )}\leq C[u]_{W^{s,p}(\Omega )}\ \ ?$$

Where $$[u]_{W^{s,p}(\Omega )}=\iint_{\Omega ^2}\frac{|u(x)-u(y)|^p}{|x-y|^{sp+n}}dxdy.$$

asked Oct 25 '17 at 12:27

idm

https://arxiv.org/pdf/1104.4345.pdf the theorem 6.5 may help. – Student Oct 25 '17 at 12:41
@Cuteboy: thanks, I know this theorem, but unfortunately, it doesn't help. – idm Oct 25 '17 at 13:28
1

It depends on the domain $\Omega$: https://arxiv.org/pdf/1705.04227.pdf – Siminore Oct 25 '17 at 14:51
@Siminore: it's perfect, thanks a lot for the link :-) – idm Oct 25 '17 at 15:04
You probably meant to write $| \cdot |_{L^p(\Omega)}$ instead of $W^{s,p}$? – Michał Miśkiewicz Oct 25 '17 at 18:43
@idm Hello, did you solve your problem? – avati91 Oct 17 '18 at 16:35
@avati91: Yes and the formula I gave is the right one. – idm Oct 17 '18 at 16:36
@idm are you familiar with fractional Sobolev spaces then? Does this inequality look true to you https://math.stackexchange.com/questions/2951520/fractional-hardy-inequality ? – avati91 Oct 17 '18 at 17:30
@avati91: Sorry, I have no idea. I wouldn't be surprised that is doesn't hold as written. But I can't find a counter example, I to be honest, I don't really know. – idm Oct 17 '18 at 17:41

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