T1 theorem on product Carnot-Caratheodory spaces

TL;DR

This paper proves a T1 theorem for product Carnot-Carathéodory spaces, extending boundedness results of certain singular integrals to product spaces with complex geometric structures.

Contribution

It establishes the product T1 theorem on $L^2$, Hardy spaces, and BMO spaces for a broad class of product singular integral operators, including those studied by Nagel, Stein, and Journé.

Findings

Proved $L^2$ boundedness of product singular integrals.

Extended Hardy space and BMO space boundedness results.

Unified framework covering previous classes of operators.

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.