Pisier's inequality revisited

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,⋯,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=1ⁿ and Δ = Σ_j=1ⁿ ∂_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 ∂_jf 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.