Boundedness of inhomogeneous Journé’s type operators on multi-parameter local Hardy spaces

Abstract

In this paper, we establish a boundedness criterion of a class of inhomogeneous Journé’s type of multi-parameter singular integral operators on multi-parameter local Hardy spaces hp(Rn1×⋯×Rnk) recently introduced in Ding et al. (2019). The consideration of such inhomogeneous Journé’s type of multi-parameter singular integral operators is motivated by the study of the multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces. The lack of the vanishing moments of both the corresponding functions in the multi-parameter local Calderón reproducing formula and the multi-parameter local Littlewood–Paley–Stein functions, as well as the lack of vanishing moments of atoms in the multi-parameter local Hardy spaces hp(Rn1×⋯×Rnk) with large size all create substantial difficulties in our proofs.