Bloom Type Upper Bounds in the Product BMO Setting

Abstract

We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral $$T_n$$ in $$\mathbb {R}^n$$ and a bounded singular integral $$T_m$$ in $$\mathbb {R}^m$$ we prove that $$\begin{aligned} \Vert [T_n^1, [b, T_m^2]] \Vert _{L^p(\mu ) \rightarrow L^p(\lambda )} \lesssim _{[\mu ]_{A_p}, [\lambda ]_{A_p}} \Vert b\Vert _{{\text {BMO}}_{prod }(\nu )}, \end{aligned}$$where $$p \in (1,\infty )$$, $$\mu , \lambda \in A_p$$ and $$\nu := \mu ^{1/p}\lambda ^{-1/p}$$ is the Bloom weight. Here $$T_n^1$$ is $$T_n$$ acting on the first variable, $$T_m^2$$ is $$T_m$$ acting on the second variable, $$A_p$$ stands for the bi-parameter weights of $$\mathbb {R}^n \times \mathbb {R}^m$$ and $${\text {BMO}}_{prod }(\nu )$$ is a weighted product BMO space.