Pisier's inequality revisited

TL;DR

This paper extends Pisier's inequality to a broader class of Banach spaces, providing uniform bounds for the associated constants and exploring conditions under which these bounds are finite.

Contribution

It generalizes Pisier's inequality to arbitrary Banach spaces and establishes conditions for uniform boundedness of the constants involved.

Findings

The constant P_p^n(X) is bounded above by the harmonic series sum for all n.

P_p^n(X) remains bounded across n if the dual space X* is UMD+ or an interpolation space.

Finite cotype Banach lattices also have uniformly bounded P_p^n(X) across n.

Abstract

Given a Banach space $X$, for $n\in \mathbb N$ and $p\in (1,\infty)$ we investigate the smallest constant $\mathfrak P\in (0,\infty)$ for which every $f_1,...,f_n:{-1,1}^n\to X$ satisfy \int_{{-1,1}^n}\Bigg|\sum_{j=1}^n \partial_jf_j(\varepsilon)\Bigg|^pd\mu(\varepsilon) \leq \mathfrak{P}^p\int_{{-1,1}^n}\int_{{-1,1}^n}\Bigg\|\sum_{j=1}^n \d_j\Delta f_j(\varepsilon)\Bigg\|^pd\mu(\varepsilon) d\mu(\delta), where $\mu$ is the uniform probability measure on the discrete hypercube ${-1,1}^n$ and ${\partial_j}_{j=1}^n$ and $\Delta=\sum_{j=1}^n\partial_j$ are the hypercube partial derivatives and the hypercube Laplacian, respectively. Denoting this constant by $\mathfrak{P}_p^n(X)$, we show that $\mathfrak{P}_p^n(X)\le \sum_{k=1}^{n}\frac{1}{k}$ for every Banach space $(X,|\cdot|)$. This extends the classical Pisier inequality, which corresponds to the special case $f_j=\Delta^{-1}\partial_j f$ for some $f:{-1,1}^n\to X$. We show that $\sup_{n\in \N}\mathfrak{P}_p^n(X)<\infty$ if either the dual $X^*$ is a $\mathrm{UMD}^+$ Banach space, or for some $\theta\in (0,1)$ we have $X=[H,Y]_\theta$, where $H$ is a Hilbert space and $Y$ is an arbitrary Banach space. It follows that $\sup_{n\in \N}\mathfrak{P}_p^n(X)<\infty$ if $X$ is a Banach lattice of finite cotype.