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

Abstract

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 Calderón--Zygmund kernels. As applications, we obtain endpoint estimates of the double Riesz transforms associated to Schrdingier operators and a Marcinkiewicz-type spectral multiplier theorem for non-negative self-adjoint operators on product spaces.

arXiv admin note: text overlap with arXiv:1002.0792, arXiv:0807.4348 by other authors

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