In Nagel et al. (J Funct Anal 181: 29--118, 2001), Nagel, Ricci and Stein established the relationships between product kernels and flag kernels on the Euclidean space. In this paper, we study the local behavior of product and flag kernels and show that local product kernels are finite sums of local flag kernels. Furthermore, we prove that the local product Hardy spaces are the intersection of local flag Hardy spaces, and local product Carleson measure spaces are the sum of local flag Carleson measure spaces.

The frist author was supported by the National Natural Science Foundation of China (Grant No. 12301115) and Natural Science Foundation of Huzhou (Grant No. 2023YZ11), and the second author was supported by the National Key Research and Development Program of China (No. 2022YFA1005703) and National Natural Science Foundation of China (No. 12071437).