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Sharp upper bound for anisotropic Rényi entropy and Heisenberg uncertainty principle

Marianna Chatzakou, Michael Ruzhansky, Anjali Shriwastawa

Abstract

In this paper, we prove the anisotropic Shannon inequality for the Renyi 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.

Sharp upper bound for anisotropic Rényi entropy and Heisenberg uncertainty principle

Abstract

In this paper, we prove the anisotropic Shannon inequality for the Renyi 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.
Paper Structure (5 sections, 6 theorems, 109 equations)

This paper contains 5 sections, 6 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Appendix
  5. Acknowledgement

Key Result

Theorem 3.1

Let $\mathbb{G}$ be a homogeneous Lie group of homogeneous dimension $Q$, and let $|\cdot|$ be a homogeneous quasi-norm on $\mathbb G$. Suppose that $\alpha>0$, $\alpha \neq 1$, and Then, for any nonnegative function $u \in L^1_b(\mathbb G)$ with $\| u\|_{L^1(\mathbb G)}=1$, the inequality holds, where and $|\mathfrak{S}|$ stands for the $Q-1$ dimensional surface measure of the unit (quasi-)sph

Theorems & Definitions (11)

  • Theorem 3.1
  • proof : Proof of Theorem \ref{['mainthm']}
  • Corollary 3.2
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Proposition 3.5
  • Theorem 3.6
  • proof
  • Corollary 3.7
  • ...and 1 more