Ultradistribution characterisation conjecture for the sub-Laplacian on the Heisenberg group

Let $\mathbb H_n$ be the Heisenberg group, let $\gamma^{s}_{\mathbf{X},L^2}(\mathbb H_n)$ and $\gamma^{(s)}_{\mathbf{X},L^2}(\mathbb H_n)$ be the corresponding sub-Laplacian Gevrey spaces, and let $u$ and $v$ be ultradistributions with group Fourier transforms $\widehat u(\lambda)$ and $\widehat v(\lambda)$. For $\lambda\in\mathbb R\setminus\{0\}$, write $\pi_\lambda(\mathcal L)$ for the infinitesimal representation of the sub-Laplacian, and let $\|\cdot\|_{\operatorname{HS}}$ denote the Hilbert–Schmidt norm. Ultradistribution characterisation conjecture. For $1\leq s<\infty$, $u\in(\gamma^{s}_{\mathbf{X},L^2}(\mathbb H_n))'$ if and only if, for every $B>0$,

$$\int_{\mathbb R\setminus\{0\}}\|e^{-B\pi_\lambda(\mathcal L)^{\frac{1}{2s}}}\widehat u(\lambda)\|_{\operatorname{HS}}|\lambda|^n\,d\lambda<\infty.$$

Moreover, $v\in(\gamma^{(s)}_{\mathbf{X},L^2}(\mathbb H_n))'$ if and only if there exists $B>0$ such that

$$\int_{\mathbb R\setminus\{0\}}\|e^{-B\pi_\lambda(\mathcal L)^{\frac{1}{2s}}}\widehat v(\lambda)\|_{\operatorname{HS}}|\lambda|^n\,d\lambda<\infty.$$

The statement is presented as an analogous Heisenberg-group characterisation based on the preceding Gevrey-space theorem; the source does not provide a proof, so its resolution remains open.