T1 theorem on product Carnot-Caratheodory spaces

2012/09/27 by Han, Yongsheng, Li, Ji, Lin, Chin-Cheng
#32W30 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 42B20 #Primary 42B35 #Secondary 32T25

paper · doi:10.48550/arxiv.1209.6236

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 \widetildeM=M₁×...× M_n, where each factor space M_i, 1≤ i≤ n, is a smooth manifold on which the basic geometry is given by a control, or Carnot--Carathéodory, metric induced by a collection of vector fields of finite type. In this paper we prove the product T1 theorem on L², the Hardy space H^p(\widetildeM) and the space CMO^p(\widetildeM), the dual of H^p(\widetildeM), for a class of product singular integral operators which covers Journé's class and operators studied by Nagel and Stein.

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