Pisier’s Inequality Revisited
نویسنده
چکیده
Given a Banach space X, for n ∈ N and p ∈ (1,∞) we investigate the smallest constant P ∈ (0,∞) for which every f1, . . . , fn : {−1, 1} → X satisfy ∫ {−1,1}n ∥∥∥∥ n ∑ j=1 ∂jfj(ε) ∥∥∥∥ p dμ(ε) 6 P ∫ {−1,1}n ∫ {−1,1}n ∥∥∥∥ n ∑ j=1 δj∆fj(ε) ∥∥∥∥ p dμ(ε)dμ(δ), where μ is the uniform probability measure on the discrete hypercube {−1, 1} and {∂j}j=1 and ∆ = ∑n j=1 ∂j are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by Pp (X), we show that P n p (X) 6 ∑n k=1 1 k for every Banach space (X, ‖ · ‖). This extends the classical Pisier inequality, which corresponds to the special case fj = ∆ ∂jf for some f : {−1, 1} → X. We show that supn∈N Pp (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 P n p (X) < ∞ if X is a Banach lattice of finite cotype.
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