Iterated commutators under a joint condition on the tuple of multiplying functions

Abstract

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

Mixed weak type inequalities in euclidean spaces and in spaces of homogeneous type

“…In the case of commutators (4.1) appeared first in [30] under some additional technical condition on A. Later on in [14,Theorem 3.5], some particular cases of that result were recovered. It is possible as well, to provide the quantitative version in terms of the sparseness constant following ideas in [11].…”

“…Section 3 is devoted to the proofs of the main results. Finally in Section 4 we show how to derive results for A-Hörmander operators and their commutators from the main results and we provide a sparse domination result for T b in the context of spaces of homogeneous type, generalizing [8,14].…”

Mixed weak‐type inequalities in Euclidean spaces and in spaces of the homogeneous type

“…In the case of commutators (4.1) appeared first in [33] under some additional technical condition on $$A$$. Later on in [16, Theorem 3.5], some particular cases of that result were recovered. It is possible as well, to provide the quantitative version in terms of the sparseness constant following ideas in [13].…”

“…Section 3 is devoted to the proofs of the main results. Finally, in Section 4 we show how to derive results for $$A$$‐Hörmander operators and their commutators from the main results and we provide a sparse domination result for $$T_{\bf b}$$ in the context of spaces of homogeneous type, generalizing [10, 16].…”

Commutators in the two scalar and matrix weighted setting

“…We will end this introduction with three remarks and an outline of the organization of the rest of the paper. First, it is an obvious question as to whether the techniques and results of this paper can be extended to the iterated commutator setting, and whether we can recover the more recent iterated commutator Bloom type results from [17] or the very recent unweighted two symbolled iterated commutator results of [8]. This will be pursued in a forthcoming paper.…”