$\mathcal{S}_w\hookrightarrow\mathcal{H}\hookrightarrow\mathcal{S}'_w$, it holds $$\mathcal{S}_w
              = \left\{T \in \mathcal{S}_w' ~\middle|~ V_e T \in \mathcal{T}_w\right\}
              = \left\{\pi(f)u ~\middle|~ f\in \mathcal{M}^{\mathcal{T}_w} \right\}.$$

Item \[2.2b\]) of the previous theorem states that the voice transform of any distribution $T\in\mathcal{S}_w'$ satisfies the reproducing formula  and uniquely defines the distribution $T$ by means of the reconstruction formula , i.e. $$T = \int_G \langle{T},{\pi(x)u}\rangle_{\mathcal{S}_w} \pi(x)u~dx,$$ where the integral is a Dunford-Pettis integral with respect to the duality between $\mathcal{S}_w$ and $\mathcal{S}_w'$, see, for example, Appendix 3 of [@fol95].

We now fix an exponent $r \in [1,\infty)$, and a $w$-moderate weight $m$, i.e. a continuous function $m : G \to (0, \infty)$ such that $$m(xy) \leq w(x) \cdot m(y)
  \quad \text{and} \quad
  m(xy) \leq m(x) \cdot w(y)
  %m(xyz) \leq w(x) m(y) w(z)
  \quad \text{for all} ~ x,y \in G \, .
  %{\color{olive} m(xyz) \leq w(x) m(y) w(z), }
  %\qquad x,y \in G \, .
  \label{eq:ModerateWeight}$$

\[rem:alt\_w-moderate\] The definition of a $w$-moderate weight $m$ is equivalent to the condition $$\begin{aligned}
m(xyz) \leq w(x)\cdot m(y)\cdot w(z) \quad \text{for all}~x,y,z\in G\end{aligned}$$ up to the constant $w(e)$.

The result of the following lemma is used multiple times in this paper.

\[lem:1/m\_w-moderate\] If $m$ is a $w$-moderate weight on $G$, then so is $m^{-1}$.

