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

## TL;DR

This paper extends Miyachi's uncertainty principle to the Laguerre hypergroup setting using the heat kernel related to the sub-Laplacian on the Heisenberg group, providing a new theoretical framework.

## Contribution

It formulates and proves a Miyachi's uncertainty principle analogue for the Laguerre hypergroup Fourier transform based on the heat kernel approach.

## Key findings

- Established an uncertainty principle for the Laguerre hypergroup.
- Connected the principle to the heat kernel of the sub-Laplacian.
- Provided a theoretical foundation for harmonic analysis on hypergroups.

## Abstract

Let $\mathbb{K}=[0,+\infty[\times\mathbb{R}$ 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.