Sobolev versus homogeneous Sobolev extension

Abstract In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results, Let $$1\leq q\leq p\leq \infty$$ 1 ≤ q ≤ p ≤ ∞ . Then a bounded $$(L^{1, p}, L^{1, q})$$ ( L 1 , p , L 1 , q ) -extension domain is also a $$(W^{1, p}, W^{1, q})$$ ( W 1 , p , W 1 , q ) -extension domain. Let $$1\leq q\leq p<q ^{\star} \leq \infty$$ 1 ≤ q ≤ p < q ⋆ ≤ ∞ or $$n< q \leq p\leq \infty$$ n < q ≤ p ≤ ∞ . Then a bounded domain is a $$(W^{1, p}, W^{1, q})$$ ( W 1 , p , W 1 , q ) -extension domain if and only if it is an $$(L^{1, p}, L^{1, q})$$ ( L 1 , p , L 1 , q ) -extension domain. For $$1\leq q<n$$ 1 ≤ q < n and $$q<nq ^{\star} <p\leq \infty$$ q < n q ⋆ < p ≤ ∞ , there exists a bounded domain $$\Omega\subset\mathbb{R}^n$$ Ω ⊂ R n which is a $$(W^{1, p}, W^{1, q})$$ ( W 1 , p , W 1 , q ) -extension domain but not an $$(L^{1, p}, L^{1, q})$$ ( L 1 , p , L 1 , q ) -extension domain for $$1 \leq q <p\leq n$$ 1 ≤ q < p ≤ n .