Hoeffding decomposition in $H^1$ spaces

Abstract

The well known result of Bourgain and Kwapień states that the projection $P_{\le m}$ onto the subspace of the Hilbert space $L^2\left({\Omega }^{\infty }\right)$ spanned by functions dependent on at most $m$ variables is bounded in $L^p$ with norm $\le c_p^m$ for $1<p<\infty$. We will be concerned with two kinds of endpoint estimates. We prove that $P_{\le m}$ is bounded on the space $H^1\left({\mathbb{D}}^{\infty }\right)$ of functions in $L^1\left({\mathbb{T}}^{\infty }\right)$ analytic in each variable. We also prove that $P_{\le 2}$ is bounded on the martingale Hardy space associated with a natural double-indexed filtration and, more generally, we exhibit a multiple indexed martingale Hardy space which contains $H^1\left({\mathbb{D}}^{\infty }\right)$ as a subspace and $P_{\le m}$ is bounded on it.