The information-theoretic meaning of Gagliardo--Nirenberg type inequalities

TL;DR

This paper explores the deep connections between Gagliardo--Nirenberg inequalities, information theory, and diffusion equations, highlighting recent advances in deriving optimal forms and explicit optimizers using entropy concepts.

Contribution

It reviews recent developments linking Gagliardo--Nirenberg inequalities with Shannon entropies and diffusion equations, emphasizing the information-theoretic perspective.

Findings

Connection between Gagliardo--Nirenberg inequalities and Shannon entropies clarified

Optimal constants and explicit optimizers derived using information-theoretic methods

Enhanced understanding of inequalities through diffusion equation analysis

Abstract

Gagliardo--Nirenberg inequalities are interpolation inequalities which were proved independently by Gagliardo and Nirenberg in the late fifties. In recent years, their connections with theoretic aspects of information theory and nonlinear diffusion equations allowed to obtain some of them in optimal form, by recovering both the sharp constants and the explicit form of the optimizers. In this note, at the light of these recent researches, we review the main connections between Shannon-type entropies, diffusion equations and a class of these inequalities.