Let $\Omega\subset\mathbb{R}^n$ be a bounded domain, with Lipschitz smooth boundary. Then a well known result by Stein gives that there exists an extension operator $E: H^k(\Omega)\rightarrow H^k(\mathbb{R}^n)$ such that
$Eu(x)=u(x)$ for all $x\in\Omega$
$\left\|Eu\right\|_{H^k(\mathbb{R}^n)}\leq C\left\|u\right\| _ {H^k(\Omega)} $
where the constant $C$ depends on $k$, $n$ and Lipschitz constant of $\Omega$. See the Thesis: A Degree-Independent Sobolev Extension Operator by Luke Rogers for a really nice summary of extension operators in the integer case.
DeVore and Sharpley in Besov spaces on bound domains of $\mathbb{R}^n$ extended this to the fractional case by real interpolation. There result is:
There exists an extension operator $E: H^\tau(\Omega)\rightarrow H^\tau(\mathbb{R}^n)$ such that:
$Eu(x)=u(x)$ for all $x\in\Omega$
$\left\|Eu\right\|_{H^\tau(\mathbb{R}^n)}\leq C\left\|u\right\| _ {H^\tau(\Omega)}$
but this time the constant depends on $\tau$, $n$ and $\Omega$. So the dependence is on $\Omega$ and not on the Lipschitz condition it satisfies.
This to me seems a highly unsatisfactory situation. Does there exist an extension operator so that the bound in the fractional Sobolev setting depends on $\Omega$ only through its Lipschitz condition? Does anyone know a reference which has this result?
