“Hoeffding decomposition in Hardy spaces.”

 We prove fact that the projection $\tilde{P}_m$ onto the subspace $\tilde{V}_m$ of $L_2\left(\mathbb{T}^\infty\right)$ 
spanned by functions dependent on at most $m$ variables 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, with norm $\leq c^m$. 
As a byproduct we get another proof the Bourgain-Kwapień theorem. We also prove that $\tilde{P}_2$ acts on 
the martingale Hardy space associated with a certain double-indexed filtration.