Kavli Affiliate: Zheng Zhu
| First 5 Authors: Pekka Koskela, Riddhi Mishra, Zheng Zhu, ,
| Summary:
In this paper, we study the relationship between Sobolev extension domains
and homogeneous Sobolev extension domains. Precisely, we obtain the following
results.
1- Let $1leq qleq pleq infty$. Then a bounded $(L^{1, p}, L^{1,
q})$-extension domain is also a $(W^{1, p}, W^{1, q})$-extension domain.
2- Let $1leq qleq p<q^starleq infty$ or $n< q leq pleq infty$. Then a
bounded domain is a $(W^{1, p}, W^{1, q})$-extension domain if and only if it
is an $(L^{1, p}, L^{1, q})$-extension domain.
3- For $1leq q<n$ and $q^star<pleq infty$, there exists a bounded domain
$Omegasubsetmathbb{R}^n$ which is a $(W^{1, p}, W^{1, q})$-extension domain
but not an $(L^{1, p}, L^{1, q})$-extension domain for $1 leq q <pleq n$.
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