Bloom type inequality for bi-parameter singular integrals: efficient proof and iterated commutators

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 $$ \| [b_k,\cdots[b_2, [b_1, T]]\cdots]\|_{L^p(\mu) \to L^p(\lambda)} \lesssim_{[\mu]_{A_p}, [\lambda]_{A_p}} \prod_{i=1}^k\|b_i\|_{\operatorname{bmo}(\nu^{\theta_i})} , $$ where $p \in (1,\infty)$, $\theta_i \in [0,1]$, $\sum_{i=1}^k\theta_i=1$, $\mu, \lambda \in A_p$, $\nu := \mu^{1/p}\lambda^{-1/p}$. Here $A_p$ stands for the bi-parameter weights in $\mathbb{R}^n \times \mathbb{R}^m$ and $\operatorname{bmo}(\nu)$ is a suitable weighted little BMO space. We also simplify the proof of the known first order case.