### 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 f_{1},⋯,f_{n}:{- 1,1}^{n} → X satisfies (Equation presented) where μ is the uniform probability measure on the discrete hypercube {-1,1}^{n}, and {∂_{j}} _{j=1}^{n} and Δ = Σ_{j=1}^{n} ∂_{j} are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by β_{p} ^{n}(X), we show that (Equation presented) for every Banach space (X, ∥ · ∥). This extends the classical Pisier inequality, which corresponds to the special case f_{j} = Δ^{-1} ∂_{j}f for some f: {- 1, 1}^{n} → X. We show that sup_{n∈ℕ βp}^{n}(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 sup_{n∈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 - Jul 1 2013 |

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### Keywords

- Enflo type
- Pisier's inequality
- Rademacher type

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Studia Mathematica*,

*215*(3), 221-235. https://doi.org/10.4064/sm215-3-2