On Hoeffding decomposition in $L_{p}$

Abstract

We give a new proof of a result by J. Bourgain which says that if $(\Omega,\mathcal{F},P)$ is a product of probability spaces then $V _d$—the orthonormal in $L_2(\Omega ,\mathcal{F},P)$ projection on the space spanned by those $X \in L_2(\Omega,\mathcal{F},P)$ which depend on most of $d$-variables is a bounded operator in $L_p(\Omega,\mathcal{F},P)$ for $1<p<\infty$. We prove that for $X \in L_p(\Omega,\mathcal{F},P)$ $E|V_d(X)|^p \le C_{p,d} E|X|^p$ with $C_{p,d}= (c\frac{\hat{p}}{\ln{\hat p}})^{dp}$, where ${\hat p}= \max\{p, \frac{p}{p-1}\}$ and $c$ is an universal constant.