We provide an explicit technical framework for proving very general twoweight commutator estimates in arbitrary parameters. The aim is to both clarify existing literature, which often explicitly focuses on two parameters only, and to extend very recent results to the full generality of arbitrary parameters. More specifically, we study two-weight commutator estimates-Bloom type estimates-in the multi-parameter setting involving weighted product BMO and little BMO spaces, and their combinations.

We complete our theory of weighted L p (w1) × L q (w2) → L r (w r/p 1 w r/q 2 ) estimates for bilinear bi-parameter Calderón-Zygmund operators under the assumption that w1 ∈ Ap and w2 ∈ Aq are bi-parameter weights. This is done by lifting a previous restriction on the class of singular integrals by extending a classical result of Muckenhoupt and Wheeden regarding weighted BMO spaces to the product BMO setting. We use this extension of the Muckenhoupt-Wheeden result also to generalise some two-weight commutator estimates from bi-parameter to multi-parameter. This gives a fully satisfactory Bloom type upper estimate for [T1, [T2, . . . [b, T k ]]], where each Ti can be a completely general multi-parameter Calderón-Zygmund operator.2010 Mathematics Subject Classification. 42B20. Key words and phrases. bilinear analysis, bi-parameter analysis, model operators, weighted estimates. H.M.

We study the bi-commutators $[T_1, [b, T_2]]$ of pointwise multiplication and Calderón–Zygmund operators and characterize their $L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}$ boundedness for several off-diagonal regimes of the mixed-norm integrability exponents $(p_1,p_2)\neq (q_1,q_2)$. The strategy is based on a bi-parameter version of the recent approximate weak factorization method.

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We develop product space theory of singular integrals with mild kernel regularity. We study these kernel regularity questions specifically in situations that are very tied to the T1 type arguments and the corresponding structural theory. In addition, our results are multilinear.