Spaces on Domains

Abstract

Up to now we dealt with spaces ${\rm{F}}^{\rm{s}} _{pq} = F^s _{pq} \left( {\mathbb{R}^n } \right),B^s _{pq} = B^s _{pq} \left( {\mathbb{R}^n } \right)$ on the one hand and their restrictions ${\rm{F}}^{\rm{s}} _{pq} \left( {\mathbb{R}_+^n } \right),B^s _{pq} \left( {\mathbb{R}_+^n } \right)$ on ${\mathbb{R}}_+^n$ on the other hand. As for the latter spaces see Definition 4.5.1. Now we introduce in precisely the same way spaces $F^s _{pq} \left( \Omega \right)$ and $B^s _{pq} \left( \Omega \right)$, where Ω always stands for a bounded C ∞ domain in ${\mathbb{R}}^n$. We are not interested in this book to deal with non-smooth domains, avoiding serious technical difficulties which may be especially hard for spaces with p<1 and/or s<0. The two outstanding problems treated here are the same as we described in 4.5.1: the extension problem and intrinsic descriptions.