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, then
∥[bk,⋯[b2,[b1,T]]⋯]∥Lp(μ)→Lp(λ)≲[μ]Ap,[λ]Api=1∏k∥bi∥bmo(νθi),
where p∈(1,∞), θi∈[0,1], ∑i=1kθi=1, μ,λ∈Ap, ν:=μ1/pλ−1/p. Here
Ap stands for the bi-parameter weights in Rn×Rm and bmo(ν) is a suitable weighted little BMO space.
We also simplify the proof of the known first order case.Juan de la Cierva - Formación 2015 FJCI-2015-24547,
BCAM Severo Ochoa excellence accreditation SEV-2013-032