# Sharp upper bound for anisotropic Rényi entropy and Heisenberg uncertainty principle

## TL;DR

This paper establishes a sharp upper bound for anisotropic Rényi entropy and derives a Heisenberg uncertainty principle on specific types of Lie groups.

## Contribution

The paper provides the best constant for the anisotropic Shannon inequality and proves an optimal Heisenberg-type uncertainty principle on stratified groups.

## Key findings

- Anisotropic Shannon inequality for Rényi entropy is proven with the best constant on Folland-Stein homogeneous Lie groups.
- A Heisenberg-type uncertainty principle is derived using a logarithmic Sobolev inequality on stratified groups.
- The optimal Shannon inequality is established in the same setting as the anisotropic inequality.

## Abstract

In this paper, we prove the anisotropic Shannon inequality for the Rényi entropy with the best constant on Folland-Stein homogeneous Lie groups. As a consequence, we also prove the optimal Shannon inequality in the same setting. Using a logarithmic Sobolev inequality in the setting of stratified groups, we prove a Heisenberg-type uncertainty principle in the latter setting.