MULTIPARAMETER HARDY SPACE THEORY ON CARNOT-CARATHEODORY SPACES AND PRODUCT SPACES OF HOMOGENEOUS TYPE

作者：Han, Yongsheng^*; Li, Ji; Lu, Guozhen

来源：Transactions of the American Mathematical Society, 2013, 365(1): 319-360.

• Kohn-Laplacian
• heat equation
• Shilov boundary
• finite type domains
• multiparameter Hardy space
• Carleson measure space
• Calderon's reproducing formula
• almost orthogonality estimate
• sup-inf-type inequality
• sequence spaces
• duality
• singular integral operators

This paper is inspired by the work of Nagel and Stein in which the L-p (1 < p < infinity) theory has been developed in the setting of the product Carnot-Caratheodory spaces (M) over tilde = M-1 x ... x M-n formed by vector fields satisfying Hormander's finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product BMO space, the boundedness of singular integral operators and Calderon-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderon identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journe-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.

• 出版日期2013-1
• 单位北京师范大学; 中山大学