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Discrete logarithmic Sobolev inequalities in Banach spaces

Dario Cordero-Erausquin, Alexandros Eskenazis

Abstract

Let $\mathscr{C}_n=\{-1,1\}^n$ be the discrete hypercube equipped with the uniform probability measure $σ_n$. We prove that if $(E,\|\cdot\|_E)$ is a Banach space of finite cotype and $p\in[1,\infty)$, then every function $f:\mathscr{C}_n\to E$ satisfies the dimension-free vector-valued $L_p$ logarithmic Sobolev inequality $$\|f-\mathbb{E} f\|_{L_p(\log L)^{p/2}(E)} \leq \mathsf{K}_p(E) \left( \int_{\mathscr{C}_n} \Big\| \sum_{i=1}^n δ_i \partial_i f\Big\|_{L_p(E)}^p \, dσ_n(δ)\right)^{1/p}.$$ The finite cotype assumption is necessary for the conclusion to hold. This estimate is the hypercube counterpart of a result of Ledoux (1988) in Gauss space and the optimal vector-valued version of a deep inequality of Talagrand (1994). As an application, we use such vector-valued $L_p$ logarithmic Sobolev inequalities to derive new lower bounds for the bi-Lipschitz distortion of nonlinear quotients of the Hamming cube into Banach spaces with prescribed Rademacher type.

Discrete logarithmic Sobolev inequalities in Banach spaces

Abstract

Let be the discrete hypercube equipped with the uniform probability measure . We prove that if is a Banach space of finite cotype and , then every function satisfies the dimension-free vector-valued logarithmic Sobolev inequality The finite cotype assumption is necessary for the conclusion to hold. This estimate is the hypercube counterpart of a result of Ledoux (1988) in Gauss space and the optimal vector-valued version of a deep inequality of Talagrand (1994). As an application, we use such vector-valued logarithmic Sobolev inequalities to derive new lower bounds for the bi-Lipschitz distortion of nonlinear quotients of the Hamming cube into Banach spaces with prescribed Rademacher type.
Paper Structure (16 sections, 15 theorems, 104 equations)

This paper contains 16 sections, 15 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Self improvement of Poincaré inequalities for hypercontractive semigroups
  3. Distortion lower bounds for quotients
  4. Preliminary discussion
  5. Historical interlude
  6. About the proof
  7. Proof of Theorem \ref{['thm:main']}
  8. Self-improvement of Poincaré inequalities
  9. Distortion lower bounds for quotients
  10. Proof of Theorem \ref{['thm:symmetric']}
  11. Orlicz norms and $\upalpha$-entropies
  12. Further remarks
  13. Limitations of semigroup methods
  14. Reductions for vector-valued influence inequalities
  15. Beckner inequalities
  16. ...and 1 more sections

Key Result

Theorem 1

Let $(E,\|\cdot\|_E)$ be a Banach space of finite cotype and $p\in[1,\infty)$. There exists a constant $\mathsf{K}_p(E)\in(0,\infty)$ such that for any $n\in\mathbb N$, every function $f:\mathscr{C}_n\to E$ satisfies

Theorems & Definitions (27)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Proposition 5: Talagrand
  • Lemma 6
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • proof : Proof of Corollary \ref{['cor:pisier']}
  • Remark 7
  • ...and 17 more