**Is there a simple proof of the following fact?**

Theorem.Let $\Omega\subset\mathbb{R}^n$ be a bounded and smooth domain. If $n>2$, then $W^{1,n-1}(\partial\Omega)\subset W^{1-\frac{1}{n},n}(\partial\Omega)$. That is, there is a bounded extension operator ${\rm Ext}:W^{1,n-1}(\partial\Omega)\to W^{1,n}(\Omega)$.

One can conclude this result from a sequence of results in H. Triebel,
*Theory of function spaces.* (Reprint of 1983 edition.)
Modern Birkhuser Classics. Birkhauser/Springer Basel AG, Basel, 2010 as follows:
using the following results Triebel's book:
Theorem 2.5.6, Theorem 2.7.1, Proposition 2.3.2.2(8), Theorem 2.5.7
and 2.5.7(9) (in that order) we obtain
the following relations for function spaces on $\mathbb{R}^{n-1}$:
$$
W^{1,n-1}(\mathbb{R}^{n-1})=
H^1_{n-1}=
F^1_{n-1,2}\subset
F^{1-\frac{1}{n}}_{n,n}=
B^{1-\frac{1}{n}}_{n,n}=
\Lambda^{1-\frac{1}{n}}_{n,n}=
W^{1-\frac{1}{n},n}(\mathbb{R}^{n-1}).
$$
I find this proof highly unsatisfactory.

A self contained and elementary (but difficult) proof can also be found in G. Leoni,
*A first course in Sobolev spaces.*
Graduate Studies in Mathematics, 105. American Mathematical Society, Providence, RI, 2009, see Theorem 14.32, Remark 14.35 and Proposition 14.40.

Non-Homogeneous Boundary Value Problems and Applications, Volume 1(Springer-Verlag, 1972), see Theorem 9.4, pages 41-43. It is still one of the best expositions on the subject. $\endgroup$ – Pedro Lauridsen Ribeiro Feb 7 at 15:39