I have been looking for (linear) Extension Operators for Slobodeckii spaces $W^{s,p}(\Omega)$ where $s>1$ and $\Omega \subset\mathbb{R}^N$ is a sufficiently smooth domain, where the influence of $\partial \Omega$ on the constants which define the continuity of this operator are given.

Indeed, by considering $E:W^{s,p}(\Omega)\to W^{s,p}(\mathbb{R}^N)$ such extension, there exists $C=C(N,s,p,\partial \Omega, \Omega)>0$ for which $$ \|E(u)\|_{W^{s,p}(\mathbb{R}^N)}\leq C \|(u)\|_{W^{s,p}(\Omega)}, \forall u \in W^{s,p}(\Omega)$$. I would like to find a reference where the dependence of $\partial \Omega$ is given explicitly.

Despite that the case $s<1$ has some good references, I can't find any good reference for $s>1$. Actually, when $s<1$, there is the nowadays classic paper by Di Nezza, Palatucci, and Valdinoci, Bull. Sci. Math. 136 (2012), no. 5, 521–573, where this kind of operator is build with certain detail. (see Thm. 5.4).

Moreover, there is also the paper of Rychov, JLMS (1999) where a universal extension operator for Besov and Triebel spaces is constructed, however, the dependence of $C$ on $\partial \Omega$ is not given.

Does anyone know any reference for this?



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