**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.



Traces of Sobolev spaces
An elementary proof was shown to me by [**Jan Malý**](https://mathscinet.ams.org/mathscinet/search/author.html?mrauthid=210365). It has never been published and with his permission we included it in [1] (Proposition 28). The argument is very elementary (1.5 pages with all details), but still too long to be included here.



> 
> **Theorem.** For $n\geq 1$ and $p>1$, there is a bounded linear extension operator $$ E:W^{1,p}(\mathbb{R}^{n})\to W^{1,q}\cap
>  C^\infty(\mathbb{R}^{n+1}\_+), \quad \text{where $q=\frac{(n+1)p}{n}$.}
>  $$ In other words, $W^{1,p}(\mathbb{R}^{n})$ continuously embeds into
>  the trace space $W^{1-\frac{1}{q},q}(\mathbb{R}^{n})$ of
>  $W^{1,q}(\mathbb{R}^{n+1}\_+)$.
> 
> 
> 


From this result the following corollary follows right away:



> 
> **Corollary.** If $\Omega\subset\mathbb{R}^n$, $n\geq 2$, is a bounded and smooth domain, then there is a bounded extension operator $$ 
>  E:W^{1,p}(\partial\Omega)\to W^{1,q}\cap C^\infty(\Omega), \quad
>  \text{where $1<p<\infty$ and $q=\frac{np}{n-1}$.} $$
> 
> 
> 


Taking $p=n-1$ and $n>2$ yields the theorem asked in the question. We need to take $n>2$ since for $n=2$ we have $p=n-1=1$ and the result is false in that case (there are counterexamples, see [1]).


**[1] P. Goldstein, P. Hajłasz,**
[Jacobians of $W^{1,p}$ homeomorphisms, case $p=[n/2]$](https://arxiv.org/abs/1812.11888). 
 *Calc. Var. Partial Differential Equations* 58 (2019), no. 4, Art. 122, 28 pp. 


