  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.
