We summarize the main properties of the extended voice transform in the following theorem.

\[intersections-1\] Under assumption , the following hold:

i)  for every $\Phi \in \mathcal{U}_w$ there exists a unique element $\pi(\Phi)u\in\mathcal{S}_w'$ such that $$\langle{\pi(\Phi)u},{v}\rangle_{\mathcal{S}_w}
                = \int_G \Phi(x) \langle{\pi(x)u},{v}\rangle_{\mathcal{H}}\,dx
                = \int_G \Phi(x) \overline{Vv(x)}\,dx,
                \qquad v \in \mathcal{S}_w.$$ Furthermore, it holds that $$V_e \pi(\Phi)u = \Phi \ast K;$$

ii) \[2.2b\] for all $T\in\mathcal{S}_w'$ the voice transform $V_eT$ is in $\mathcal{U}_w$ and satisfies $$\begin{aligned}
              & V_e T = V_e T \ast K, \label{eq:ExtendedVoiceTransformReproducing} \\
              & \langle{T},{v}\rangle_{\mathcal{S}_w} = \langle{V_e T},{V v}\rangle_w,
              \qquad v \in \mathcal{S}_w;
              \label{eq:ExtendedVoiceTransformDuality}
            \end{aligned}$$

iii) the extended voice transform $V_e$ is injective, continuous from $\mathcal{S}_w'$ into $\mathcal{U}_w$ (when both spaces are endowed with the strong topology), its range is the closed subspace $$\mathcal M^{\mathcal{U}_w}
              = \left\{\Phi \in \mathcal{U}_w ~\middle|~ \Phi \ast K = \Phi\right\}
              = \operatorname{span}
                  \bigcup_{p \in (1,\infty)}
                    \mathcal{M}^{L_{p,w}(G)}
              \subset L_{\infty,{w^{-1}}}(G)
              \label{eq:38}$$ and it intertwines the contragradient representation of $\pi_{|\mathcal{S}_w}$ and $\lambda_{|\mathcal{U}_w}$;

iv) \[sotto\] the map $$\mathcal M^{\mathcal{U}_w} \ni \Phi \mapsto \pi(\Phi) u \in \mathcal{S}_w'$$ is the left inverse of $V_e$ and coincides with the restriction of the map $\,^t\!{\,V}$ to $\mathcal M^{\mathcal{U}_w}$, namely $$V_e( \,^t\!{\,V}\Phi)
              = V_e\pi(\Phi)u
              = \Phi,
              \qquad \Phi \in \mathcal M^{\mathcal{U}_w};
              \label{eq:39}$$

v)  regarding 