2025

Koskela, P.; Mishra, R.; Zhu, Z.

In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results. center dot Let 1 <= q <= p <= infinity. Then a bounded (L-1 ,L-p, L-1 ,L-q)-extension domain is also a (W-1 ,W-p, W-1 ,W-q)-extension domain. center dot Let 1 <= q <= p < q(star) < infinity or n < q <= p <= infinity. Then abounded domain is a (W-1 ,W-p, W-1 ,W-q)-extension domain if and only if it is an (L-1 ,L-p, L-1 ,L-q)-extension domain. center dot For 1 <= q < n and q(star) < p <= infinity, there exists a bounded domain ohm subset of R-n which is a (W-1 ,W-p, W-1 ,W-q)-extension domain but not an (L-1 ,L-p, L-1 ,L-q)-extension domain for 1 <= q < p <= n.