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We defined the Weighted Sobolev semi-norm by $$[u]_{W^{s,p,\alpha }(\Omega )}^p=\iint_{\Omega \times \Omega }\frac{|u(x)-u(y)|^p |x|^{\alpha _1p}|y|^{\alpha _2p}}{|x-y|^{sp+d}}dxdy,$$ where $\alpha =\alpha _1+\alpha _2$, where $p\in(1,\infty )$. The thing I don't understand is why the case $p=1$ is not allowed ?

You can see reference here or here. I really don't understand why $p=1$ can't be considered.

idm
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  • I certainly have answer for this these are fractional Sobolev spaces right? – Guy Fsone Feb 08 '18 at 10:02
  • Just a personal question are you work in a project right now? because you have some very tedious questions that need time to answer – Guy Fsone Feb 08 '18 at 10:03
  • @GuyFsone: Kind of. I also have this question in reference of the paper you put on the bounty I gave you here : https://math.stackexchange.com/questions/2640062/gagliardo-niremberg-inequality-on-annuli-proof It's based on [3] Gagliardo : ulteriori proprietà di alcune classi di funzioni in più variabili, I'm sure the answer is in this paper, but I can't find it. Do you know where I can find this paper ? – idm Feb 08 '18 at 10:21
  • I will try to see whether I can prove it – Guy Fsone Feb 08 '18 at 15:57
  • what are the restriction on $\alpha?$ I know already that $0<s<1$ – Guy Fsone Feb 08 '18 at 20:53
  • @GuyFsone: as we can imagine : that $[u]_{W^{s,p,\alpha }}<\infty $ – idm Feb 08 '18 at 21:04
  • My question is ? $\alpha $ is in which range? can we take it arbitrally real? I doubt. – Guy Fsone Feb 08 '18 at 21:50
  • @GuyFsone: I follow this paper : https://www.sciencedirect.com/science/article/pii/S0022123617302732 and there is no specific restriction. – idm Feb 09 '18 at 08:05

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