On the product theory of singular integrals

Abstract

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.