Spaces on domains: localization and Hardy inequalities

Abstract

Let Ω be a domain in ${\mathbb{R}^n}$. Then the spaces $B_{pq}^s\left( \Omega \right)$ and $F_{pq}^s\left( \Omega \right)$ can be intro­duced by restriction of $B_{pq}^s\left( {{\mathbb{R}^n}} \right)$ and $F_{pq}^s\left( {{\mathbb{R}^n}} \right)$ on Ω, respectively. This applies in particular to the special cases considered in 1.2: Sobolev spaces (classical and fractional), Hölder-Zygmund spaces, classical Besov spaces etc. These concrete spaces, but also the general scales $B_{pq}^s\left( \Omega \right)$ and $F_{pq}^s\left( \Omega \right)$, have been studied in great detail. We refer to the books listed in 1.1. It is the aim of this section and the following one to contribute to this theory in the spirit of Sections2 and 3, where we discussed corresponding spaces on ${\mathbb{R}^n}$.