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, 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.

  • $\begingroup$ For $L^2$ Sobolev spaces, a reasonably elementary Fourier-analytic proof may be found in J.L. Lions and E. Magenes, 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
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    $\begingroup$ @PedroLauridsenRibeiro Theroem 9.4 is about characterization of traces in terms of fractional Sobolev spaces. $W^{1,n-1}$ is not a fractional Sobolev space and the only question is to show that it embeds to a suitable fractional Sobolev space. Moreover the Theorem 9.4 applies to when $p=2$ only. Note that here $p-n>2$. Therefore Theorem 9.4 is not relevant here. But thank you for the reference. I will certainly look at it more carefully. Looks like a great book that I wan to add to my library. $\endgroup$ – Piotr Hajlasz Feb 7 at 16:20

An elementary proof was shown to me by Jan Malý. 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 $p=[n/2]$. (Preprint 2018).


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