Criterion of the L2 boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type

The purpose of this paper is to introduce a class of general singular integral operators on spaces M = M1 × ......... × Mn. Each factor space Mi 1 < i <n, is a space of homogeneous type in the sense of Coifman and Weiss. These operators generalize those studied by Journe on the Euclidean space and include operators studied by Nagel and Stein on Carnot-Caratheodory spaces on which the basic geometry is given by a control, or Carnot-Caratheodory, metric induced by a collection of vector fields of finite type. We provide the criterion of the L2(M) boundedness for these general operators. Thus this result extends the product T theorem of Journe on Euclidean space and recovers the Lp, 1 < p < ∞, boundedness of those operators on Carnot-Caratheodory space obtained by Nagel and Stein. We also prove the sharp endpoint estimates for these general operators on the Hardy spaces Hp(M) and BMO(M).