A heat kernel version of Miyachi's Theorem for the Laguerre hypergroup

Let $\mathbb{K}=[0,+\mathrm{\infty }[\times \mathbb{R}$ the Laguerre Hypergroup. In this paper, we are going to formulate and prove an analogue of Miyachi’s uncertainty principle for the Laguerre-Hypergroup Fourier transform. Our version will be in terms of the heat kernel associated to the radial part of the sub-Laplacian on the Heisenberg group.