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. 1- Let $1\le q\le p\le \infty$. Then a bounded $(L^{1,p},L^{1,q})$-extension domain is also a $(W^{1,p},W^{1,q})$-extension domain. 2- Let $1\le q\le p<q^{\star }\le \infty$ or $n<q\le p\le \infty$. 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 $1\le q<n$ and $q^{\star }<p\le \infty$, there exists a bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ which is a $(W^{1,p},W^{1,q})$-extension domain but not an $(L^{1,p},L^{1,q})$-extension domain for $1\le q<p\le n$.