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I already ask the question on MSE (here), but I didn't had any answer, so I try here.

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

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  • $\begingroup$ I don't think the issue is that they cannot define the seminorm; I think the issue is that the results/method of proof in the papers only applies to give good inequalities when $p >1$. (Where did you see in the papers that say the seminorm cannot be defined when $p = 1$? For the unweighted case certainly the classical Slobodeckij seminorm allows $p = 1$.) $\endgroup$
    – Willie Wong
    Commented Feb 7, 2018 at 16:39
  • $\begingroup$ @WillieWong : Thank you for your answer. The writer of the second paper (H.-M. Nguyen) told me that the double integral may not converge when $p=1$. But to me it also may not converge when $p\neq 1$, no ? $\endgroup$
    – idm
    Commented Feb 7, 2018 at 18:14

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