Anisotropic singular integrals in product spaces

In this paper, the authors introduce a class of product anisotropic singular integral operators, whose kernels are adapted to the action of a pair $ \vec A $ := (A ₁, A ₂) of expansive dilations on ? ⁿ and ? ^m , respectively. This class is a generalization of product singular integrals with convolution kernels introduced in the isotropic setting by Fefferman and Stein. The authors establish the boundedness of these operators in weighted Lebesgue and Hardy spaces with weights in product A _∞ Muckenhoupt weights on ? ⁿ × ? ^m . These results are new even in the unweighted setting for product anisotropic Hardy spaces.