# Revista Matemática Iberoamericana

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**Volume 20, Issue 2, 2004, pp. 531–561**

**DOI: 10.4171/RMI/400**

Published online: 2004-08-31

On the product theory of singular integrals

Alexander Nagel^{[1]}and Elias M. Stein

^{[2]}(1) University of Wisconsin, Madison, USA

(2) Princeton University, United States

We establish $L^p$-boundedness for a class of product singular integral operators on spaces $\widetilde{M} = M_1 \times M_2\times \cdots \times M_n$. Each factor space $M_i$ is a smooth manifold on which the basic geometry is given by a control, or Carnot-Caratheodory, metric induced by a collection of vector fields of finite type. The standard singular integrals on $M_i$ are non-isotropic smoothing operators of order zero. The boundedness of the product operators is then a consequence of a natural Littlewood-Paley theory on $\widetilde M$. This in turn is a consequence of a corresponding theory on each factor space. The square function for this theory is constructed from the heat kernel for the sub-Laplacian on each factor.

*Keywords: *Product singular integrals, control metrics, NIS operators, Littlewood-Paley theory

Nagel Alexander, Stein Elias: On the product theory of singular integrals. *Rev. Mat. Iberoam.* 20 (2004), 531-561. doi: 10.4171/RMI/400