Structured, compactly supported Banach frame decompositions of decomposition spaces

Understand

$\newcommand{mc}[1]{\mathcal{#1}}$ $\newcommand{D}{\mc{D}(\mc{Q},L^p,\ell_w^q)}$ We present a framework for the construction of structured, possibly compactly supported Banach frames and atomic decompositions for decomposition spaces.

• Such a space $\D$ is defined using a frequency covering $\mc{Q}=(Q_i)_{i\in I}$: If $(\varphi_i)_{i}$ is a suitable partition of unity subordinate to $\mc{Q}$, then $\Vert g\Vert_{\D}:=\left\Vert\left(\Vert\mc{F}^{-1}(\varphi_i\hat{g})\Vert_{L^p}\right)_{i}\right\Vert_{\ell_w^q}$.
• We assume $\mc{Q}=(T_iQ+b_i)_{i}$, with $T_i\in{\rm GL}(\Bbb{R}^d),b_i\in\Bbb{R}^d$.
• Given a prototype $\gamma$, we consider the system \[\Psi_{c}=(L_{c\cdot T_i^{-T}k}\gamma^{[i]})_{i\in I,k\in\Bbb{Z}^d}\text{ with }\gamma^{[i]}=|\det T_i|^{1/2}\, M_{b_i}(\gamma\circ T_i^T),\] with translation $L_x$ and modulation $M_{\xi}$.