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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.

EDIT: There is a simple (unpublished) proof due to Jan Malý. I will write it when I have time.

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.

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.

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.

EDIT: There is a simple (unpublished) proof due to Jan Malý. I will write it when I have time.

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.

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.