We establish -boundedness for a class of product singular integral operators on spaces . Each factor space 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 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 . 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.
Cite this article
Alexander Nagel, Elias M. Stein, On the product theory of singular integrals. Rev. Mat. Iberoam. 20 (2004), no. 2, pp. 531–561DOI 10.4171/RMI/400