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Dyadic bi-parameter repeated commutator and dyadic product BMO

Consider a tensor product of simple dyadic shifts defined below. We prove here that for dyadic bi-parameter repeated commutator its norm can be estimated from below by Chang-Fefferman $BMO$ norm pertinent to its symbol. See Theorems in Section 8 at the end of this article. But this is done below under an extra assumption on the Haar--Fourier side of the symbol. In Section 7 we carefully analyze what goes wrong in the absence of this extra assumption. At the end of this note we also list a counterexample to the existing proof of characterization of bi-parameter repeated commutator with the Hilbert transforms. This is a counterexample to the proof, and it is not a counterexample to the statement of factorization result in bi-disc, or to Nehari's theorem in bi-disc. To the best of our knowledge Nehari's theorem on bi-disc is still open. Moreover its dyadic bi-parameter version considered in the present paper is also still open for general symbol without any extra restrictions.