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An information-theoretic proof of a hypercontractive inequality

Ehud Friedgut

Abstract

In this note I give an information-theoretic proof of the Bonami-Beckner-Gross hypercontractive inequality.

An information-theoretic proof of a hypercontractive inequality

Abstract

In this note I give an information-theoretic proof of the Bonami-Beckner-Gross hypercontractive inequality.

Paper Structure

This paper contains 4 sections, 3 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Main Theorem
  3. The Boolean case
  4. The general (non-Boolean) case

Key Result

Theorem 1.1

: [Gross,Bonami,Beckner] Let $f : \{0,1\}^n \rightarrow \mathbb{R}^{\ge 0}$, and let $\epsilon \in [0,1]$. Then where, as usual, $|g|_p = E[|g|^p]$, and the expectation is with respect to the uniform measure.

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 2.1
  • Claim 2.2
  • Theorem 2.3