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• # Artykuł - szczegóły

## Studia Mathematica

2013 | 215 | 3 | 221-235

## Pisier's inequality revisited

EN

### Abstrakty

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

221-235

wydano
2013

### Twórcy

autor
• Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FI-00014 Helsinki, Finland
autor
• Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, U.S.A.