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=1n ∂j 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-1 ∂jf 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 language | English (US) |
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Pages (from-to) | 221-235 |
Number of pages | 15 |
Journal | Studia Mathematica |
Volume | 215 |
Issue number | 3 |
DOIs | |
State | Published - 2013 |
Keywords
- Enflo type
- Pisier's inequality
- Rademacher type
ASJC Scopus subject areas
- General Mathematics