Bloom Type Upper Bounds in the Product BMO Setting

We prove some Bloom type estimates in the product BMO setting. More specifically, for a bounded singular integral T n in R n and a bounded singular integral T m in R m we prove that ‖ [ T n 1 , [ b , T m 2 ] ] ‖ L p ( μ ) → L p ( λ ) ≲ [ μ ] A p , [ λ ] A p ‖ b ‖ BMO prod ( ν ) , where p ∈ ( 1 , ∞ ) , μ , λ ∈ A p and ν : = μ 1 / p λ - 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 R n × R m and BMO prod ( ν ) is a weighted product BMO space.