Bloom type upper bounds in the product BMO setting

TL;DR

This paper establishes Bloom-type upper bounds for commutators of bounded singular integrals in a bi-parameter setting, linking operator norms to a weighted product BMO space, advancing understanding in multi-parameter harmonic analysis.

Contribution

It proves new bounds for commutators of bi-parameter singular integrals with weights, extending Bloom-type results to the product BMO setting.

Findings

Boundedness of commutators in weighted L^p spaces.

Extension of Bloom bounds to product BMO spaces.

Quantitative estimates depending on A_p weights.

Abstract

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 $$ \| [T_n^1, [b, T_m^2]] \|_{L^p(\mu) \to L^p(\lambda)} \lesssim_{[\mu]_{A_p}, [\lambda]_{A_p}} \|b\|_{\operatorname{BMO}_{\textrm{prod}}(\nu)}, $$ where $p \in (1,\infty)$, $\mu, \lambda \in A_p$ and $\nu := \mu^{1/p}\lambda^{-1/p}$. 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 $\operatorname{BMO}_{\textrm{prod}}(\nu)$ is a weighted product BMO space.