$F \in \mathcal{M}_{r,m}$.

Since $\mathcal{M}_{r,m}$ is a closed subspace of $L_{r,m}(G)$, [@Mo00 Proposition 1.4] yields that $\mathcal{M}'_{r,m}$ is canonically isomorphic to $L'_{r,m}(G)/\mathcal{M}_{r,m}^\bot$. The claim follows because $L'_{r,m}(G)$ is canonically isomorphic to $L_{r',m^{-1}}(G)$.

Some more preparations are necessary. Given a compact neighborhood $Q \subset G$ of $e$ with $Q = \overline{\operatorname{int} Q}$, the *local maximal function (with respect to the right regular representation)* $M^\rho_Q f$ of $f \in  L_0(G) $ is defined by $$\begin{aligned}
\label{local_maximum_function}
  M^\rho_Q f(x) := \|f \cdot \rho(x)\chi_Q\|_{L_\infty},
  \quad \text{whence} \quad
  \widecheck{M}^\rho_Q f(x) := M^\rho_Q f (x^{-1}) = \| f \|_{L_\infty(Q x)} \, .\end{aligned}$$ Then, for a function space $Y$ on $G$, we define $$\begin{aligned}
\label{wiener_amalgam_space}
  \mathcal{M}^\rho_Q(Y)
  := \left\{
        f \in  L_0(G) 
        ~\middle|~
        \vphantom{M^j} \smash{\widecheck{M}^\rho_Q} f \in Y
     \right\}.\end{aligned}$$

Now we define the [*$Q$-oscillation*]{} of a function $f$ with respect to $Q$ as $$\label{oscillation}
  \mbox{osc}_Q f(x) := \sup_{u \in Q} |f(ux)-f(x)|.$$ The decay-properties of the $Q$-oscillation play an important role in view of the discretization of coorbit spaces. To this end, the following lemma is useful. Since the proof is a simple generalization of the proof of [@gro91 Lemma 4.6], it is deferred to the appendix.

\[lem:osc\] Let $w$ be a weight on $G$, let $p\in(1,\infty)$, and assume 