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.