# Traces of Sobolev spaces

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.

• 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. Feb 7, 2019 at 15:39
• @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. Feb 7, 2019 at 16:20

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 $$W^{1,q}(\mathbb{R}^{n+1}_+)$$.
Corollary. If $$\Omega\subset\mathbb{R}^n$$, $$n\geq 2$$, is a bounded and smooth domain, then there is a bounded extension operator $$E:W^{1,p}(\partial\Omega)\to W^{1,q}\cap C^\infty(\Omega), \quad \text{where 1
Taking $$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$$ and the result is false in that case (there are counterexamples, see ).
 P. Goldstein, P. Hajłasz, Jacobians of $$W^{1,p}$$ homeomorphisms, case $$p=[n/2]$$. Calc. Var. Partial Differential Equations 58 (2019), no. 4, Art. 122, 28 pp.