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, then $$\Vert [b_k,\cdots [b_2,[b_1,T]]\cdots ]{\Vert }_{L^p(\mu )\to L^p(\lambda )}{\lesssim }_{[\mu {]}_{A_p},[\lambda {]}_{A_p}}\prod _{i=1}^k\Vert b_i{\Vert }_{\mathrm{bmo}({\nu }^{{\theta }_i})},$$ 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 $\mathrm{bmo}(\nu )$ is a suitable weighted little BMO space. We also simplify the proof of the known first order case.

v3: Incorporated referee comments, to appear in Int. Math. Res. Not. IMRN

References in corpus (5)

• The L^p-to-L^q boundedness of commutators with applications to the Jacobian operator
• Bilinear Calderón-Zygmund theory on product spaces
• Sparse Domination for Bi-Parameter Operators Using Square Functions
• Two-weight inequalities for multilinear commutators
• Commutators of bilinear bi-parameter singular integrals