Multi-parameter local Hardy spaces. (July 2019) - Ding, Wei; Lu, Guozhen; Zhu, YuePing
Abstract: Though multi-parameter Hardy space theory has been well developed in the past half century, not much has been studied for a local Hardy space theory in the multi-parameter settings. Such multi-parameter local Hardy spaces can play an important role in studying the boundedness of multi-parameter pseudo-differential operators, multi-parameter singular integrals of non-convolution type, and applications to partial differential equations, etc. By establishing a bi-parameter local reproducing formula, bi-parameter local Hardy space h p ( R n 1 × R n 2 ) is introduced in this paper. This space coincides with L p ( R n 1 + n 2 ) when p > 1 . While p ≤ 1, this space is substantially different from the classical bi-parameter Hardy spaces H p ( R n 1 × R n 2 ) . We will establish its atomic decomposition of the bi-parameter local Hardy spaces and as an application, we prove the boundedness from h p ( R n 1 × R n 2 ) to L p ( R n 1 + n 2 ) of the bi-parameter singular integral operators in inhomogeneous Journé class for p near 1. For the simplicity, we have chosen to present all the results in the bi-parameter setting. Nevertheless, all of them hold for arbitrary number of parameters. The multi-parameter local theory developed in this paper can serve as a model case for similar theory in other multi-parameter settings.