Pisier's inequality revisited

Tuomas Hytönen, Assaf Naor

Research output: Contribution to journalArticle

Abstract

Given a Banach space X, for n ∈ N and p ∈ (1,∞) we investigate the smallest constant β ∈ (0, ∞) for which every n-tuple of functions f1,⋯,fn:{- 1,1}n → X satisfies (Equation presented) where μ is the uniform probability measure on the discrete hypercube {-1,1}n, and {∂j} j=1n and Δ = Σj=1nj are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by βp n(X), we show that (Equation presented) for every Banach space (X, ∥ · ∥). This extends the classical Pisier inequality, which corresponds to the special case fj = Δ-1jf for some f: {- 1, 1}n → X. We show that supn∈ℕ βpn(X) < ∞ if either the dual X* is a UMD Banach space, or for some θ ∈ (0,1) we have X = [H,Y]θ, where H is a Hilbert space and Y is an arbitrary Banach space. It follows that supn∈ℕ βp n(X) < ∞ if X is a Banach lattice of nontrivial type.

Original languageEnglish (US)
Pages (from-to)221-235
Number of pages15
JournalStudia Mathematica
Volume215
Issue number3
DOIs
StatePublished - Jul 1 2013

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Keywords

  • Enflo type
  • Pisier's inequality
  • Rademacher type

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Hytönen, T., & Naor, A. (2013). Pisier's inequality revisited. Studia Mathematica, 215(3), 221-235. https://doi.org/10.4064/sm215-3-2