Independent sums of H¹_n(mathbb{T}) and H¹_n(δ)

We construct a new idempotent Fourier multiplier on the Hardy space on the bidisc, which could not be obtained by applying known one dimentional results. The main tool is a new $L^1$ equivalent of the Stein martingale inequality which holds for a special filtration of periodic subsets of $\mathbb{T}$ 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 $H^1_n$, which is known to be a complemented and invariant subspace of dyadic $H^1$.