2012/07/23 by Tuomas Hytönen, Hytönen, Tuomas, Assaf Naor +1
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.FA
paper · pdf · doi:10.48550/arxiv.1207.5375
Referee comments addressed. To appear in Studia Mathematica
openalex publication_date 2012/07/23 · arxiv created 2013/05/14 · arxiv updated 2013/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a Banach space X, for n∈ \mathbb N and p∈ (1,∞) we investigate the smallest constant \mathfrak P∈ (0,∞) for which every f1,...,fn:-1,1n→ X satisfy ∫_-1,1n|∑j=1n ∂jfj(ε)|pdμ(ε) ≤ \mathfrakPp∫_-1,1n∫_-1,1n‖∑j=1n \djΔfj(ε)‖pdμ(ε) dμ(δ), where μ is the uniform probability measure on the discrete hypercube -1,1n and ∂jj=1n and Δ=∑j=1n∂j are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by \mathfrakPpn(X), we show that \mathfrakPpn(X)≤ ∑k=1n(1)/(k) for every Banach space (X,|⋅|). This extends the classical Pisier inequality, which corresponds to the special case fj=Δ-1∂j f for some f:-1,1n→ X. We show that supn∈ \N\mathfrakPpn(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\mathfrakPpn(X)<∞ if X is a Banach lattice of finite cotype.