Pisier's inequality revisited

Tuomas Hytönen, and Assaf Naor. "Pisier's inequality revisited." Studia Mathematica 215.3 (2013): 221-235. <http://eudml.org/doc/285609>.

@article{TuomasHytönen2013,
abstract = {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.},
author = {Tuomas Hytönen, Assaf Naor},
journal = {Studia Mathematica},
keywords = {Enflo type; Pisier's inequality; Rademacher type; unconditionality for martingale differences},
language = {eng},
number = {3},
pages = {221-235},
title = {Pisier's inequality revisited},
url = {http://eudml.org/doc/285609},
volume = {215},
year = {2013},
}

TY - JOUR
AU - Tuomas Hytönen
AU - Assaf Naor
TI - Pisier's inequality revisited
JO - Studia Mathematica
PY - 2013
VL - 215
IS - 3
SP - 221
EP - 235
AB - 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.
LA - eng
KW - Enflo type; Pisier's inequality; Rademacher type; unconditionality for martingale differences
UR - http://eudml.org/doc/285609
ER -