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.
We present a pair of joint conditions on the two functions b 1 , b 2 strictly weaker than b 1 , b 2 ∈ BMO that almost characterize the L 2 boundedness of the iterated commutator [b 2 , [b 1 , T ]] of these functions and a Calderón-Zygmund operator T. Namely, we sandwich this boundedness between two bisublinear mean oscillation conditions of which one is a slightly bumped up version of the other.2010 Mathematics Subject Classification. 42B20.Before proceeding any further, let us precisely define the commutatorwhere [A, B] = AB − BA for any two operations A, B, and b i f (·) = b i (·)f (·).We deal with the second order commutator [b 2 , [b 1 , T ]] but our results concerning sufficient conditions could just as well be formulated in the higher order cases.It follows by the John-Nirenberg inequality that if b 1 , b 2 ∈ BMO, then the conditions S p , T q hold for all p, q ≥ 1. Hence, a natural question is immediate: Are S p , and respectively T p , equivalent for all 1 ≤ p < ∞. Or even in a weaker sense: if both of the conditions S p , T p hold simultaneously, could we deduce S q or T q for some q > p? By Theorem 2.4 the answer is no.The next proposition will clarify the situation and point out how the counterexample in Theorem 2.4 can be constructed.Recall, that a function ω : R d → (0, ∞) is said to be in the class of A p weights, 1 < p < ∞, if[w] Ap = sup Q w Q
The authors were supported by the Academy of Finland through project Nos. 314829 (all), 336323 (Sinko) and 346314 (Hytönen), as well as by the Jenny and Antti Wihuri Foundation (Oikari).1 For notation see Section 2.2 below.
By extending the approximate weak factorization method to the bilinear setting we identify testing conditions on the function b that characterize the L p × L q → L r boundedness of the commutatorfor many exponents in the range 1 < p, q < ∞ and r > 1 2 , where T is a bilinear Calderón-Zygmund operator.2010 Mathematics Subject Classification. 42B20.
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