Criterion of the L-2 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 M̃ = 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 Journé on the Euclidean space and include operators studied by Nagel and Stein on Carnot-Carathéodory, metric induced by a collection of vector fields of finite type. We provide the criterion of the L²(M̃) boundedness for these general operators. Thus this result extends the product T 1 theorem of Journé on Euclidean space and recovers the Lᴾ₊ 1 <p <∞, boundedness of those operators on Carnot-Carathéodory space obtained by Nagel and Stein. We also prove the sharp endpoint estimates for these general operators on the Hardy spaces Hᴾ(M̃) and BMO(M̃).