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

TL;DR

The paper addresses establishing sharp discrete Gagliardo-Nirenberg and logarithmic Sobolev inequalities on the integer lattice $\mathbb{Z}^n$ and characterizing the rigidity of extremals. It introduces a discrete Brascamp-Lieb inequality with a complete rigidity description, showing equality occurs only for $f=\lambda\cdot\chi_A$ with $A=\prod_{i=1}^n \pi_i(A)$. Using this, it derives the sharp lattice Gagliardo-Nirenberg inequality $\|f\|_{\frac{n}{n-1}} \le \frac{1}{2} \prod_{i=1}^n \|\partial_i f\|_1^{1/n}$ and the sharp lattice Sobolev inequality $\|f\|_{\frac{n}{n-1}} \le \frac{1}{2n} \|df\|_1$, with equalities attained on cuboids and cubes respectively. Additionally, the paper obtains sharp logarithmic Sobolev inequalities on $\mathbb{Z}^n$ and relates these results to discrete variants of Loomis-Whitney and isoperimetric inequalities.

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.

Theorems & Definitions (10)

• Theorem 1.1
• Corollary 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.2
• Corollary 1.4
• proof : Proof of Theorem \ref{['thm-BL']}
• proof : Proof of Corollary \ref{['cor-log-BL']}
• proof : Proof of Theorem \ref{['thm-G-N']}
• proof : Proof of Corollary \ref{['cor-Sobolev']}