$P$-log-Sobolev inequalities on $\mathbb{N}$

TL;DR

This paper investigates the relationships between different p-log-Sobolev inequalities on the natural numbers, demonstrating that for any interval within (0,1], certain inequalities can hold while others fail, revealing nuanced distinctions.

Contribution

It provides the first examples showing that p-log-Sobolev inequalities can differ within (0,1], and develops new criteria for these inequalities in birth-death processes.

Findings

Existence of measures where q-log-Sobolev holds but p-log-Sobolev fails for q<p in (0,1].

Development of necessary and sufficient conditions for p-log-Sobolev inequalities on irth-death processes.

Resolution of an open problem posed by Mossel, Oleszkiewicz, and Sen.

Abstract

We answer an open problem posed by Mossel--Oleszkiewicz--Sen regarding relations between $p$-log-Sobolev inequalities for $p\in(0,1]$. We show that for any interval $I\subset(0,1]$, there exist $q,p\in I$, $q<p$, and a measure $\mu$ for which the $q$-log-Sobolev inequality holds, while the $p$-log-Sobolev inequality is violated. As a tool we develop certain necessary and closely related sufficient conditions characterizing those inequalities in the case of birth-death processes on $\mathbb{N}$.