Pisier's inequality revisited

Tuomas Hytönen¹ ; Assaf Naor²

¹ Department of Mathematics and Statistics University of Helsinki P.O. Box 68 FI-00014 Helsinki, Finland
² Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, NY 10012, U.S.A.

Studia Mathematica, Tome 215 (2013) no. 3, pp. 221-235

Résumé

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 $n$-tuple of functions $f_1,\ldots,f_n\!:\!\{-1,1\}^n\!\to\! X$ satisfies \[\def\e{\varepsilon}\def\d{\delta}\int_{\{-1,1\}^n}\Big\|\sum_{j=1}^n \partial_jf_j(\e)\Big\|^p\,d\mu(\varepsilon)\le \mathfrak{P}^p\int_{\{-1,1\}^n}\int_{\{-1,1\}^n}\Big\|\sum_{j=1}^n \d_j\varDelta f_j(\varepsilon)\Big\|^p\,d\mu(\varepsilon) \,d\mu(\delta), \] where $\mu$ is the uniform probability measure on the discrete hypercube $\{-1,1\}^n$, and $\{\partial_j\}_{j=1}^n$ and $\varDelta=\sum_{j=1}^n\partial_j$ 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 $f_j=\varDelta^{-1}\partial_j f$ 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 $X^*$ 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 nontrivial type.

Détail

DOI : 10.4064/sm215-3-2

Keywords: given banach space mathbb infty investigate smallest constant mathfrak infty which every n tuple functions ldots satisfies def varepsilon def delta int sum partial varepsilon mathfrak int int sum vardelta varepsilon varepsilon delta where uniform probability measure discrete hypercube partial vardelta sum partial hypercube partial derivatives hypercube laplacian respectively denoting constant mathfrak mathfrak sum frac every banach space cdot extends classical pisier inequality which corresponds special vardelta partial sup mathbb mathfrak infty either dual * mathrm umd banach space theta have theta where hilbert space arbitrary banach space follows sup mathbb mathfrak infty banach lattice nontrivial type