that $w^{-1} \not\in L_q(G)$ for all $1 < q < \infty$.

We now define the *test space* $\mathcal{S}_w$ as $$\mathcal{S}_w = \left\{
                v\in\mathcal{H}
                ~\middle|~
                Vv \in L_{p,w}(G) \text{ for all } 1 < p < \infty
              \right\},
  \label{eq:43}$$ which becomes a locally convex topological vector space under the family of semi-norms $$\lVert{v}\rVert_{p,\mathcal{S}_w} = \lVert{Vv}\rVert_{L_{p,w}}.
  \label{eq:29}$$ We recall the main properties of $\mathcal{S}_w$.

\[intersections\] Under Assumption , the following hold:

i)  the space $\mathcal{S}_w$ is a reflexive Fréchet space, continuously and densely embedded in $\mathcal{H}$;

ii) \[2.1b\] the representation $\pi$ leaves $\mathcal{S}_w$ invariant and its restriction to $\mathcal{S}_w$ is a continuous representation;

iii) the space $\mathcal{H}$ is continuously and densely embedded into the (anti)-linear dual $\mathcal{S}'_w$, where both spaces are endowed with the weak topology;

iv) the restriction of the voice transform $V:\mathcal{S}_w\to \mathcal{T}_w$ is a topological isomorphism from $\mathcal{S}_w$ onto the closed subspace $\mathcal{M}^{\mathcal{T}_w}$ of $\mathcal{T}_w$, given by $$\mathcal M^{\mathcal{T}_w} = \left\{f\in \mathcal{T}_w ~\middle|~ f \ast K = f\right\},$$ and it intertwines $\pi$ and $\lambda$;

v)  \[bar\] for every $f\in \mathcal{T}_w$, there exists a unique element $\pi(f)u\in\mathcal{S}_w$ such that $$\langle{\pi(f)u},{v}\rangle_\mathcal{H}
                = \int_G f(x) \langle{\pi(x)u},{v}\rangle_\mathcal{H}~dx
                = \int_G f(x) \overline{Vv(x)}~dx,
                \qquad v\in\mathcal{H}.$$ Furthermore, it holds that $$V\pi(f)u = f \ast K,$$ and 