# Hoeffding decomposition in $H^1$ spaces

## TL;DR

This paper investigates the boundedness of projection operators in various Hardy spaces, extending known results to endpoint cases in $H^1$ spaces and martingale Hardy spaces, with implications for harmonic analysis.

## Contribution

It proves boundedness of projections in $H^1$ and martingale Hardy spaces, extending Bourgain and Kwapień's results to endpoint and analytic function settings.

## Key findings

- Boundedness of $P_{ ext{leq } m}$ in $H^1( ext{D}^ ext{infty})$
- Boundedness of $P_{ ext{leq } 2}$ in martingale Hardy spaces
- Introduction of a multiple indexed martingale Hardy space containing $H^1( ext{D}^ extinfty)$

## Abstract

The well known result of Bourgain and Kwapie\'n states that the projection $P_{\leq m}$ 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 $\leq c_p^m$ for $1<p<\infty$. We will be concerned with two kinds of endpoint estimates. We prove that $P_{\leq m}$ 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 $P_{\leq 2}$ 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 $P_{\leq m}$ is bounded on it.