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.