Poincaré type inequalities for group measure spaces and related transportation cost inequalities

Qiang Zeng*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

Abstract

Let G be a countable discrete group with an orthogonal representation α on a real Hilbert space H. We prove Lp Poincaré inequalities for the group measure space LH, γ)⋊G, where both the group action and the Gaussian measure space (ΩH, γ) are associated with the representation α. The idea of proof comes from Pisier's method on the boundedness of Riesz transform and Lust-Piquard's work on spin systems. Then we deduce a transportation type inequality from the Lp Poincaré inequalities in the general noncommutative setting. This inequality is sharp up to a constant (in the Gaussian setting). Several applications are given, including Wiener/Rademacher chaos estimation and new examples of Rieffel's compact quantum metric spaces.

Original languageEnglish (US)
Pages (from-to)3236-3264
Number of pages29
JournalJournal of Functional Analysis
Volume266
Issue number5
DOIs
StatePublished - Mar 1 2014

Keywords

  • Group measure spaces
  • Poincaré type inequalities
  • Quantum metric spaces
  • Rademacher chaos
  • Transportation cost inequalities
  • Wiener chaos

ASJC Scopus subject areas

  • Analysis

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