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It is known that the fractional $L^p$ Sobolev inequality $$ \|f\|_{L^{p^*_s}(\mathbb R^n)}^p \leq \sigma_{n,p,s} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy $$ with $p^*_s = \frac{np}{n-sp}$, $p \in (1, n/s)$, holds for every smooth function with compact support in $\mathbb R^n$.

(see for example Theorem 1 in Maz’ya, V.; Shaposhnikova, T., On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195, No. 2, 230-238 (2002). ZBL1028.46050.)

Let's assume that $\sigma_{n,p,s}$ here is the best possible constant.

It is also known that $$ \lim_{s \to 1^-} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy = \alpha_{n,p} \| \nabla f \|_{L^p{(\mathbb R^n)}}^p $$ where $\alpha_{n,p}$ is a constant depending only on $n$ and $p$.

(see, Bourgain, Jean; Brezis, Haim; Mironescu, Petru, Another look at Sobolev spaces, Menaldi, José Luis (ed.) et al., Optimal control and partial differential equations. In honour of Professor Alain Bensoussan’s 60th birthday. Proceedings of the conference, Paris, France, December 4, 2000. Amsterdam: IOS Press; Tokyo: Ohmsha (ISBN 1-58603-096-5; 4-274-90412-1). 439-455 (2001). ZBL1103.46310.)

Finally, the $L^p$ Sobolev inequality $$ \|f\|_{L^{p^*}(\mathbb R^n)}^p \leq \sigma_{n,p} \| \nabla f \|_{L^p{(\mathbb R^n)}}^p, $$ with $p^* = \frac{np}{n-p}$, $p \in (1,n)$, holds for every smooth function with compact support in $\mathbb R^n$.

Assume $\sigma_{n,p}$ is the best possible constant.

The values of $\sigma_{n,p}$ and $\alpha_{n,p}$ are explicitly known but, to my knowledge, the values of $\sigma_{n,p,s}$ are not.

Is it true that $\lim_{s \to 1^-} \sigma_{n,p,s} = \sigma_{n,p} \alpha_{n,p}^{-1}$ ?

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