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

Bartłomiej Polaczyk

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 $μ$ 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 $\mathbb{N}$.

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

Abstract

We answer an open problem posed by Mossel--Oleszkiewicz--Sen regarding relations between -log-Sobolev inequalities for . We show that for any interval , there exist , , and a measure for which the -log-Sobolev inequality holds, while the -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 .
Paper Structure (13 sections, 15 theorems, 110 equations)

This paper contains 13 sections, 15 theorems, 110 equations.

Table of Contents

  1. Preliminaries
  2. Background
  3. General setup
  4. Main result
  5. Auxiliary results
  6. Hardy inequality
  7. Tail estimates
  8. Reduction to increasing functions
  9. Technical lemmas
  10. Proof of Theorem \ref{['T:main']}
  11. Sufficient condition
  12. Necessary condition
  13. Proof of Theorem \ref{['C:main']}

Key Result

Proposition 1.2

The Poincaré inequality eq:Poincare with constant $C>0$ is equivalent to the 0-log-Sobolev inequality with the same constant $C$.

Theorems & Definitions (30)

  • Definition 1.1
  • Proposition 1.2: MosselOleszkiewiczSen
  • Proposition 1.3: MosselOleszkiewiczSen
  • Proposition 1.4: MosselOleszkiewiczSen
  • Remark 1.5
  • Theorem 1.6: MosselOleszkiewiczSen
  • Remark 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • ...and 20 more