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Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Suppose that there there is a bounded extension operator $$ E:W^{1,p}(\Omega)\to W^{1,p}(\mathbb{R}^n) \quad \text{and} \quad E:W^{1,q}(\Omega)\to W^{1,q}(\mathbb{R}^n), $$ where $1\leq p<q\leq\infty$ (or $1<p<q<\infty$ if you find it easier).

Open problem. Does it follow that $$ E:W^{1,r}(\Omega)\to W^{1,r}(\mathbb{R}^n) $$ for all $p<r<q$?

Some partial results are known, but the general case seems to be difficult. It looks like a simple interpolation exercise, but it is not.

Later, when I have time I will add some references to known results.

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  • $\begingroup$ I'm taking the risk of stepping into the "simple interpolation exercise"-trap: What is keeping us from using the interpolation result for Sobolev spaces on domains as in Liu/Tai Theorem 9 and reiterate to verify that $E$ is a continuous linear operator between $W^{1,r}(\Omega)$ and $W^{1,r}(\mathbb{R}^n)$? (Then verify the extension/restriction property by hand or abstractly using Triebel's retraction/coretraction theorem..) $\endgroup$ – Hannes Dec 2 '18 at 15:19
  • $\begingroup$ @Hannes Theorem 9 in Liu-Tai works only for nice domains like extension domains and we do not know that our domain is the extension one. For example, it is not known if for general domains $W^{1,p}(\Omega)$ is isomorphic to $L^p$. For extension domains there is such an isomorphism by Pelczynski's theorem. $\endgroup$ – Piotr Hajlasz Dec 2 '18 at 15:35
  • $\begingroup$ Well, so far I had thought that the remarkable thing about the Liu/Tai paper was that it did NOT need additional domain/boundary regularity.. could you point me to the place where there are (maybe also implicit) assumption, even if it then does not help your question? $\endgroup$ – Hannes Dec 2 '18 at 15:50
  • $\begingroup$ Good point. They seem to prove the result for general domains, but I am pretty sure that it is not easy since the K-interpolation argument is well known and I know that many people tried to approach to the case of general domain without much success. I have to read the paper by Liu-Tai. Thank you for pointing it out to me. $\endgroup$ – Piotr Hajlasz Dec 2 '18 at 16:34
  • $\begingroup$ I also became aware of the paper only recently and was quite surprised, especially because it is already 20 years old. Glad if it helps, I hope there's nothing problematic hidden somewhere. $\endgroup$ – Hannes Dec 2 '18 at 17:32

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