End-point estimates for singular integrals with non-smooth kernels on product spaces

The main aim of this article is to establish boundedness of singular integrals with non-smooth kernels on product spaces. Let $L_1$ and $L_2$ be non-negative self-adjoint operators on $L^2({\mathbb{R}}^{n_1})$ and $L^2({\mathbb{R}}^{n_2})$, respectively, whose heat kernels satisfy Gaussian upper bounds. First, we obtain an atomic decomposition for functions in $H_{L_1,L_2}^1({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})$ where the Hardy space $H_{L_1,L_2}^1({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})$ associated with $L_1$ and $L_2$ is defined by square function norms, then prove an interpolation property for this space. Next, we establish sufficient conditions for certain singular integral operators to be bounded on the Hardy space $H_{L_1,L_2}^1({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})$ when the associated kernels of these singular integrals only satisfy regularity conditions significantly weaker than those of the standard Calder'on–Zygmund kernels. As applications, we obtain endpoint estimates of the double Riesz transforms associated to Schr"dingier operators and a Marcinkiewicz-type spectral multiplier theorem for non-negative self-adjoint operators on product spaces.