Sobolev extensions, interpolation inequalities and consequences

We prove Sobolev interpolation inequalities on extension domains that have a form reminiscent of the corresponding whole-space inequalities. This form is crucial in certain applications, which we discuss as well. The technical key ingredient is the notion of a Lebesgue $W^{1,p}$-extension domain, which we introduce here, and our proof that, for $1<p<\mathrm{\infty }$, any $W^{1,p}$-extension domain is a Lebesgue $W^{1,p}$-extension domain.