Bloom-Type Inequality for Bi-Parameter Singular Integrals: Efficient Proof and Iterated Commutators

Abstract

Utilising some recent ideas from our bilinear bi-parameter theory, we give an efficient proof of a two-weight Bloom type inequality for iterated commutators of linear bi-parameter singular integrals. We prove that if T is a bi-parameter singular integral satisfying the assumptions of the bi-parameter representation theorem, thenHere Ap stands for the bi-parameter weights in R n × R m and bmo(ν) is a suitable weighted little BMO space. We also simplify the proof of the known first order case.2010 Mathematics Subject Classification. 42B20.

Bloom Type Upper Bounds in the Product BMO Setting

“…The product BMO Bloom type upper bound is the main contribution of this paper. However, we also complement our previous paper [33] and (1.1) by giving an easy little BMO iterated commutator lower bound proof using the so-called median method, previously used in the one-parameter setting in [31,22] (see also [32]). In [33] we only recorded the following remark regarding the Estimate (1.1).…”

“…However, we also complement our previous paper [33] and (1.1) by giving an easy little BMO iterated commutator lower bound proof using the so-called median method, previously used in the one-parameter setting in [31,22] (see also [32]). In [33] we only recorded the following remark regarding the Estimate (1.1). Choosing b 1 = · · · = b k = b and θ 1 = · · · = θ k = 1/k in (1.1) we get a bi-parameter analog of [31], while choosing θ 1 = 1 (and the rest zero) we get analogs of [17,21].…”

“…Here BMO prod (ν) is a weighted product BMO space as defined at least in [16,33]. We note that bmo(ν) ⊂ BMO prod (ν) (a proof can be found at least in [33]). This product BMO Bloom estimate is proved in Section 4.…”

“…This product BMO Bloom estimate is proved in Section 4. While we were inspired by our previous paper [33] dealing with the new approach to little BMO commutators, a quite different take on things is required by the current product BMO setting. The proof idea is outlined in the beginning of Section 4.…”

“…In the recent paper [33] we reproved the result of [16] in an efficient way based on improved commutator decompositions from our bilinear bi-parameter theory [32]. The clear structure of our proof allowed us to also handle the iterated little BMO commutator case and to prove the upper bound…”