Pisier's inequality revisited

Tuomas Hytönen ; Assaf Naor

Studia Mathematica, Tome 215 (2013), p. 221-235 / Harvested from The Polish Digital Mathematics Library

Résumé

Given a Banach space X, for n ∈ ℕ and p ∈ (1,∞) we investigate the smallest constant ∈ (0,∞) for which every n-tuple of functions f₁,...,fₙ: -1,1ⁿ → X satisfies $∫_{{-1,1}ⁿ} ||∑_{j=1}^{n} ∂_{j}f_{j}(ε)||^{p} dμ(ε) ≤ ^{p} ∫_{{-1,1}ⁿ} ∫_{{-1,1}ⁿ} ||∑_{j=1}^{n} δ_{j} Δf_{j}(ε)||^{p} dμ(ε)dμ(δ)$, where μ is the uniform probability measure on the discrete hypercube -1,1ⁿ, and ${∂_j}_{j=1}^{n}$ and $Δ = ∑_{j=1}^{n}∂_{j}$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by $ⁿ_{p}(X)$, we show that $ⁿ_{p}(X) ≤ ∑_{k=1}^{n} 1/k$ 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: -1,1ⁿ → X. We show that $sup_{n∈ ℕ }ⁿ_{p}(X) < ∞$ 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 $sup_{n∈ ℕ}ⁿ_{p}(X) < ∞$ if X is a Banach lattice of nontrivial type.

Détail

Publié le : 2013-01-01

Zbl 1285.46007

EUDML-ID : urn:eudml:doc:285609