What question did this study set out to answer?

The study aims to establish the anisotropic Shannon inequality and an uncertainty principle based on Rényi entropy.

March 16, 2026manuscripta mathematica0 citationsOpen Access

Sharp upper bound for anisotropic Rényi entropy and Heisenberg uncertainty principle

MCMarianna ChatzakouUniversity College Ghent MRMichael RuzhanskyQueen Mary University of London ASAnjali ShriwastawaBanaras Hindu University

Key Points

The study aims to establish the anisotropic Shannon inequality and an uncertainty principle based on Rényi entropy.
Proved anisotropic Shannon inequality for Rényi entropy on Folland-Stein homogeneous Lie groups.
Derived the optimal Shannon inequality in the same framework.
Utilized a logarithmic Sobolev inequality in stratified groups for further proofs.
Established best constant for anisotropic Shannon inequality.
Demonstrated optimal Shannon inequality performance.
Proved Heisenberg-type uncertainty principle in the context of stratified groups.

Abstract

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.

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