Journal of the
Korean Mathematical Society JKMS

In this paper, we study the boundedness of a class of inhomogeneous Journ\'{e}'s product singular integral operators on the inhomogeneous product Lipschitz spaces. The consideration of such inhomogeneous Journ\'{e}'s product singular integral operators is motivated by the study of the multi-parameter pseudo-differential operators. The key idea used here is to develop the Littlewood-Paley theory for the inhomogeneous product spaces which includes the characterization of a special inhomogeneous product Besov space and a density argument for the inhomogeneous product Lipschitz spaces in the weak sense.

Keywords: Calder\'{o}n-Zygmund operator, inhomogeneous product Lipschitz space, Littlewood-Paley theory

MSC numbers: Primary 42B20; Secondary 42B25, 46E30

Supported by: The first author was supported by Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ22A010018) and the second author was supported by National Natural Science Foundation of China (Grant Nos. 11626213, 11771399).