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We construct a new idempotent Fourier multiplier on the Hardy space on thebidisc, which could not be obtained by applying known one dimentional results.The main tool is a new $L^1$ equivalent of the Stein martingale inequalitywhich holds for a special filtration of periodic subsets of $\mathbb{T}$ withsome restrictions on the functions involved. We also identify the isomorphictype 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$H^1$.

Independent sums of $H^1_n(\mathbb{T})$ and $H^1_n(\delta)$