To prove the estimates in for $m^{-1}$ we fix $x,y\in G$, then by the $w$-moderateness of $m$ it holds $$\begin{aligned}
m(y) = m(x^{-1}xy) \leq w(x^{-1}) \cdot m(xy) = w(x) \cdot m(xy),\end{aligned}$$ which implies $m(xy)^{-1}\leq w(x) \cdot m(y)^{-1}$. Similarly we observe that $$\begin{aligned}
m(x) = m(xyy^{-1}) \leq m(xy) \cdot w(y^{-1}) = m(xy) \cdot w(y),\end{aligned}$$ which in turn implies $m(xy)^{-1}\leq m(x)^{-1} \cdot w(y)$.

With terminology as in [@dadedelastte17], we choose as a [*model space*]{} for the coorbit space theory, the Banach space $Y = L_{r,m}(G)$ with $r\in (1,\infty)$. The corresponding coorbit space is defined as $$\operatorname{Co}(Y) = \left\{T\in \mathcal{S}'_w ~\middle|~ V_e T \in Y \right\}
 \label{eq:21}$$ endowed with the norm $$\lVert{T}\rVert_{\operatorname{Co}(Y)} = \lVert{V_eT}\rVert_Y.
 \label{eq:23}$$ We summarize the main properties of $\operatorname{Co}(Y)$ in the following proposition.

\[prop6\] The space $\operatorname{Co}(Y)$ is a Banach space invariant under the action of the contragradient representation of $\pi_{|\mathcal{S}_w}$. The extended voice transform is an isometry from $\operatorname{Co}(Y)$ onto the $\lambda$-invariant closed subspace $$\mathcal M^Y = \left\{F\in Y ~\middle|~ F \ast  K =F \right\} \subset \mathcal{U}_w \, ,$$ and we have $$\operatorname{Co}(Y) = \left\{\pi(F)u ~\middle|~ F \in \mathcal{M}^Y \right\}.$$ Furthermore $$\begin{aligned}
 V_e \pi(F)u & = F,
 \qquad F \in \mathcal{M}^Y,
 \label{eq:65} \\
 \pi(V_eT)u & = T,
 \qquad T 