Bloom type upper bounds in the product BMO setting

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 $$\Vert [T_n^1,[b,T_m^2]]{\Vert }_{L^p(\mu )\to L^p(\lambda )}{\lesssim }_{[\mu {]}_{A_p},[\lambda {]}_{A_p}}\Vert b{\Vert }_{{\mathrm{BMO}}_{\text{prod}}(\nu )},$$ 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 ${\mathrm{BMO}}_{\text{prod}}(\nu )$ is a weighted product BMO space.

This version to appear in J. Geom. Anal.; 18 pages

References in corpus (3)

• Bilinear Calderón-Zygmund theory on product spaces
• Sparse Domination for Bi-Parameter Operators Using Square Functions
• Two-weight inequalities for multilinear commutators