T1 theorem on product Carnot-Caratheodory spaces

Abstract

Nagel and Stein established $L^p$-boundedness for a class of singular integrals of NIS type, that is, non-isotropic smoothing operators of order 0, on spaces $\widetilde{M}=M_1\times...\times M_n,$ where each factor space $M_i, 1\leq i\leq n,$ is a smooth manifold on which the basic geometry is given by a control, or Carnot--Carath\'eodory, metric induced by a collection of vector fields of finite type. In this paper we prove the product $T1$ theorem on $L^2,$ the Hardy space $H^p(\widetilde{M})$ and the space $CMO^p(\widetilde{M})$, the dual of $H^p(\widetilde{M}),$ for a class of product singular integral operators which covers Journ\'e's class and operators studied by Nagel and Stein.