Hypercontractive Inequalities for the Second Norm of Highly Concentrated Functions, and Mrs. Gerber’s-Type Inequalities for the Second Rényi Entropy

Niv Levhari, Alex Samorodnitsky

Research output: Contribution to journalArticlepeer-review

Abstract

Let (Formula presented.), (Formula presented.), be the noise operator acting on functions on the boolean cube (Formula presented.). Let f be a distribution on (Formula presented.) and let (Formula presented.). We prove tight Mrs. Gerber-type results for the second Rényi entropy of (Formula presented.) which take into account the value of the (Formula presented.) Rényi entropy of f. For a general function f on (Formula presented.) we prove tight hypercontractive inequalities for the (Formula presented.) norm of (Formula presented.) which take into account the ratio between (Formula presented.) and (Formula presented.) norms of f.

Original languageEnglish
Article number1376
JournalEntropy
Volume24
Issue number10
DOIs
StatePublished - Oct 2022
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2022 by the authors.

Keywords

  • Mrs. Gerber’s inequality
  • Rényi entropy
  • entropy
  • hypercontractivity

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