is to establish certain atomic decompositions for the coorbit spaces described in Sect. \[sec:overview\]. In particular, we recall that $$\label{basicsetting}
  {\mathcal T}_{w} = \bigcap_{p \in (1,\infty)} L_{p,w}(G),
  \quad\quad
  {\mathcal T}'_{w}
  = {\mathcal U}_{w}
  = \mbox{span} \bigcup_{q \in (1,\infty)} L_{q,w^{-1}}(G)$$ and for some $1<r<\infty$, $$\label{Lmspaces}
  Y = L_{r,m}(G).$$ Proposition \[prop6\] shows that the correspondence principle holds, i.e., the extended voice transform $V_e$ is an isometry from the associated coorbit space $$\label{defcoorbit}
  \mbox{Co}( L_{r,m})
  := \left\{ T \in {\mathcal S}'_{w} ~\middle|~ V_e(T) \in L_{r,m}(G) \right\}$$ onto the corresponding reproducing kernel Banach space $$\label{eq:ReproducingKernelLp}
  \mathcal{M}_{r,m} = \mathcal{M}^{L_{r,m}(G)} = \left\{ f \in L_{r,m}(G) ~\middle|~ f \ast K = f \right\}.$$

Assumption  on the kernel $K$ and the fact that $m$ is $w$-moderate imply that for all $f\in L_{r,m}(G)$ the convolution $f\ast K$ is well-defined; see Proposition \[prop:Young\].

In this setting we can characterize the anti-dual $\mathcal{M}'_{r,m}$ of the reproducing kernel space.

\[dualreproducingkernel\] The anti-dual $\mathcal{M}'_{r,m}$ of $\mathcal{M}_{r,m}$ is canonically isomorphic to $L_{r',m^{-1}}(G)/\mathcal{M}_{r,m}^\bot$, where $$\begin{aligned}
  \mathcal{M}_{r,m}^\bot
  = \left\{
      \widetilde{F} \in L_{r',m^{-1}}(G)
      ~\middle|~
      \langle{\widetilde{F} },{F}\rangle_{L_2} = 0
      \mbox{ for all } F \in \mathcal{M}_{r,m}
    \right\}
  \label{eq:ReproducingKernelSpaceDual}\end{aligned}$$ and $1/r+1/r'=1$. Hence, for every $\Gamma\in\mathcal{M}'_{r,m}$ there is a $\widetilde{F} \in L_{r',m^{-1}}(G)$ such that $\Gamma(F) = \langle{\widetilde{F} },{F}\rangle_{L_2}$ for all 