Spaces on domains: decompositions

Abstract

Let Ω be a bounded ${C^\infty }$ domain in ${\mathbb{R}^n}$, let {φ_jr} be the resolution of unity according to 5.13, and letpqsbe restricted by (5.119). Then by Theorem 5.14 any $f \in \tilde F_{pq}^s\left( \Omega \right)$ can be (almost obviously) represented as

$$ f\sum\limits_{j,r} {{\varphi _{jr}}\left( x \right)} f\left( {convergence in {L_1}\left( \Omega \right)} \right), $$

(6.1)

and (not so obviously) (5.120) is an equivalent quasi-norm. It is the main aim of this section to combine this refined localization assertion with quarkonial decompositions, now applied to φ_jr f, according to Definition 2.6 and Theorem 2.9. Although clear in principle, the details cause some technical problems.Recall that according to 5.22, with exception of some singular cases, the above spaces $\tilde F_{pq}^s\left( \Omega \right)$ coincide with $\mathop{{F_{{pq}}^{s}}}\limits^{ \circ } \left( \Omega \right),$ and can be described by (5.166).