Pisier's inequality revisited

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|^pdμ(\varepsilon) \leq \mathfrak{P}^p\int_{{-1,1}^n}\int_{{-1,1}^n}\Bigg\|\sum_{j=1}^n \d_jΔf_j(\varepsilon)\Bigg\|^pdμ(\varepsilon) dμ(δ), 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 ${\sup }_{n\in \mathbb{N}}{\mathfrak{P}}_p^n(X)<\infty$ if either the dual $X^{\ast }$ 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 ${\sup }_{n\in \mathbb{N}}{\mathfrak{P}}_p^n(X)<\infty$ if $X$ is a Banach lattice of finite cotype.

arXiv commentReferee comments addressed. To appear in Studia Mathematica