Sharp Gagliardo-Nirenberg inequality and logarithmic Sobolev inequality on integer lattices

TL;DR

This paper establishes sharp Gagliardo-Nirenberg and logarithmic Sobolev inequalities on integer lattices, providing precise bounds and characterizations that advance understanding of functional inequalities in discrete settings.

Contribution

The paper introduces the first sharp Gagliardo-Nirenberg inequality on integer lattices and derives associated sharp logarithmic Sobolev inequalities, including rigidity characterizations.

Findings

Sharp Gagliardo-Nirenberg inequality on integer lattices

Rigidity characterization of the inequality

Sharp logarithmic Sobolev inequalities derived

Abstract

In this paper, we obtain a sharp Garliardo-Nirenberg inequality on integer lattices and characterize its rigidity. Moreover, as a consequence of the sharp Garliardo-Nirenberg inequality, we obtain sharp logarithmic Sobolev inequalities on integer lattices.