Let $\Omega \subset \mathbb{R}^d$ be a domain with Lipschitz boundary and $W_p^k(\Omega)$ be the Sobolev space of functions over $\Omega$ for which weak derivatives up to order $k$ exist and have finite a $L^p$ norm.
I am interested in the existence of extension operators from $W_p^k(\Omega)$ to $W_p^k(\mathbb{R}^d)$. I have found reference for the case when $k$ is an integer (see [1]) and for when $k \in (0,1)$ (see [2]). I assume that the results also hold for any $k$ of the form $k = n + s$ where $s \in [0,1)$ and $n$ is a non-negative integer. Can anyone point out a reference which has the proof for this general case? I believe it is based on combining ideas from the two references but I can't figure out the exact details myself and it would be much easier for me to just cite the appropriate reference if it has already been proven.
Thanks for your help.
[1] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions", Princeton University Press, 1970. ISBN 978069108079
[2] F. Demengel and G. Demengel, "Fractional Sobolev Spaces", pages 179–228. Springer London, London, 2012. ISBN 978-1-4471-2807-6.