Sobolev Versus Homogeneous Sobolev II

We study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, for a certain range of exponents $p$ and $q$, we construct a $(W^{1,p},W^{1,q})$-extension domain which is not an $(L^{1,p},L^{1,q})$-extension domain.