Bloom type upper bounds in the product BMO setting (1810.09303v2)

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.