Paley–Wiener conjecture for the Dunkl transform

Let $G$ be a finite reflection group. For a subset $S\subset\mathfrak a$, let ${\mathcal D}(S)$ denote the smooth compactly supported functions with support contained in $S$. For a non-empty compact subset $S$, define the indicator

$$I_S(x)=\max_{y\in S}(x,y),\qquad x\in\mathfrak a.$$

Let ${\mathcal H}_S$ be the space of entire functions on $\mathfrak a_{\mathbb C}$ such that, for every integer $M\geq 0$, there exists a constant $\gamma_M$ satisfying

$$|f(\lambda)|\leq \gamma_M(1+|\lambda|)^{-M}\exp I_S(\operatorname{Im}\lambda)$$

for all $\lambda\in\mathfrak a_{\mathbb C}$. Paley–Wiener conjecture. If $\operatorname{Re} k\geq 0$ and $S$ is a non-empty $G$-invariant compact convex subset of $\mathfrak a$, then $D_k$ is a linear isomorphism between ${\mathcal D}(S)$ and ${\mathcal H}_S$. This would give a geometric Paley–Wiener theorem for the Dunkl transform; the paper presents several supporting theorems, but the stated isomorphism is left as a conjecture.