Quarkonial analysis

The spaces $A_{p,q}^s({\mathbb{R}}^n)$ with $A\in \{B,F\}$, $s\in \mathbb{R}$ and $0<p,q\le \mathrm{\infty }$ are usually introduced in terms of Fourier–analytical decompositions. Related characterizations based on atoms and wavelets are known nowadays in a rather final way. Quarks atomize the atoms into constructive building blocks. It is the main aim of these notes to raise quarkonial decompositions to the same level as related representations of the spaces $A_{p,q}^s({\mathbb{R}}^n)$ in terms of atoms or wavelets. This will be complemented by some applications. In addition we deal also with quarks in domains and their relations to so–called refined localization spaces.