# Uncertainty principles on nilpotent Lie groups

## TL;DR

This paper extends classical uncertainty principles like Hardy's and Beurling's theorems to the setting of connected nilpotent Lie groups, analyzing Fourier and Gabor transforms.

## Contribution

It proves Hardy's uncertainty principle and an analogue of Hardy's theorem for Gabor transforms on connected nilpotent Lie groups, and discusses Beurling's theorem for specific group products.

## Key findings

- Hardy's uncertainty principle is established for Fourier transforms on nilpotent Lie groups.
- An analogue of Hardy's theorem for Gabor transform is proved for these groups.
- Beurling's theorem for Gabor transform is discussed for groups of the form R_n × K.

## Abstract

Hardy's type uncertainty principle on connected nilpotent Lie groups for the Fourier transform is proved. An analogue of Hardy's theorem for Gabor transform has been established for connected and simply connected nilpotent Lie groups. Finally Beurling's theorem for Gabor transform is discussed for groups of the form $\mathbb{R}_n \times K$, where $K$ is a compact group