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. $\endgroup$ Feb 7 '19 at 15:39
  • 1
    $\begingroup$ @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. $\endgroup$ Feb 7 '19 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 $p=[n/2]$. Calc. Var. Partial Differential Equations 58 (2019), no. 4, Art. 122, 28 pp.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.