two functions $f, g \in L_0(G)$, with slight abuse of notations, we write $$\begin{aligned}
  \langle{f},{g}\rangle_{L_2} = \int_G f(x)\overline{g(x)}\,dx,\end{aligned}$$ provided that the function $fg$ is integrable.

We fix a continuous weight $w : G \to (0, \infty)$ satisfying

$$\begin{aligned}
    w(xy) & \leq w(x)w(y), \label{eq:ControlWeightSubmultiplicative} \\
    w(x) & = w(x^{-1})    \label{eq:ControlWeightSymmetric}
  \end{aligned}$$

for all $x,y \in G$. As a consequence, it also holds that $$\inf_{x\in G} w(x) \geq 1 \label{eq:ControlWeightBoundedBelow}.$$

The symmetry  can always be satisfied by replacing $w$ with $w+\widecheck{w}$, where the latter weight is easily seen to still satisfy the submultiplicativity condition . For all $p\in[1,\infty)$ define the separable Banach space $$L_{p,w}(G) = \left\{f\in  L_0(G)  ~\middle|~ \int_G \lvert{w(x)f(x)}\rvert^p ~dx<\infty\right\}$$ with norm $$\lVert f \rVert_{L_{{p,w}}}^p = \int_G \lvert{w(x)f(x)}\rvert^p ~dx,$$ and the obvious modifications for $L_\infty(G)$, which however is not separable. When $w \equiv 1$ we simply write $L_p(G)$.

With terminology as in [@dadedelastte17] we choose, as a [*target space*]{} for the coorbit space theory, the following space $$\mathcal{T}_w =\bigcap_{1<p<\infty} L_{p,w}(G).$$ We recall some basic properties of $\mathcal{T}_w$; for proofs we refer to Theorem 4.3 of [@dadedelastte17], which is based on results in [@damuwe70]. We endow $\mathcal{T}_w$ with the (unique) topology such that a sequence $(f_n)_{\in\mathbb{N}}$ in $\mathcal{T}_w$ converges to $0$ if 