This is an announcement for the paper "Pisier's inequality revisited" by Tuomas Hytonen and Assaf Naor.
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 $f_1,\ldots,f_n:{-1,1}^n\to X$ satisfy \begin{multline*} \int_{{-1,1}^n}\Bigg|\sum_{j=1}^n \partial_jf_j(\varepsilon)\Bigg|^pd\mu(\varepsilon)\\le \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), \end{multline*} 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.
Archive classification: math.FA
Submitted from: naor@cims.nyu.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1207.5375
or