Independent sums of Hn1(T) and Hn1(δ)

We construct a new idempotent Fourier multiplier on the Hardy space on the bidisc, which could not be obtained by applying known one dimensional results. The main tool is a new L1 equivalent of the Stein martingale inequality which holds for a special filtration of periodic subsets of T with some restrictions on the functions involved. We also identify the isomorphic type of the range of the associated operator as the independent sum of dyadic Hn1, which is known to be a complemented and invariant subspace of dyadic H1.