Multi-parameter Hardy Spaces And Dual Spaces Associated With Flag Structure And Singular Integrals

In this dissertation,we focus on the study of multi-parameter Hardy spaces,du-ality and the boundedness of the singular integrals and mainly consider the following four questions:three-parameter Hardy space associated with a sum of two flag singular integrals and its duality and the boundedness of multi-parameter singular integrals on the above two spaces;the weighted multi-parameter local Hardy spaces K?p of arbitrary k number of parameters and boundedness of singular integrals of convolution type on this space,where the weight belongs to A? class and k ?3;the boundedness of Journé's type singular integrals on the multi-parameter Lipschitz spaces,including the product Lipschitz spaces,the inhomogeneous product spaces,the bi-parameter mixed Lipschitz spaces;the boundedness of multidimensional Hausdorff operators on Hp(Rn)(0<p<1)and Lp(Rn)(p>1).The dissertation is divided into seven chapters.In Charter 1,we introduce the background and main results in the dissertation.In Chapter 2,we establish the three-parameter Hardy space associated with a sum of two flag singular integrals and its dual space and the boundedness of multi-parameter singular integrals on the above two spaces,and characterize this Hardy space as the intersection of flag Hardy spaces and characterize its dual space as the sum of flag Carleson measure spaces.The key idea used here is the discrete Littlewood-Paley-Stein theory.In Charter 3,we apply the discrete multi-parameter local Calderon identity and Littlewood-Paley-Stein theory with weights to carry out the weighted multi-parameter local Hardy spaces h?p of arbitrary k number of parameters(k? 3)and derive the h?p boundedness for multi-parameter singular integral operators with slightly weaker assump-tions on the kernel.In Charter 4,we develop the Littlewood-Paley theory for the product Lipschitz spaces and establish a necessary and sufficient condition for the boundedness of product Calderon-Zygmund singular integral operators introduced by Journé.In Charter 5,we establish a boundedness criterion of a class of singular integral operators on the inhomogeneous product Lipschitz spaces,including the multi-parameter pseudo-differential operators and inhomogeneous Journé's product singular integral.In Charter 6,we introduce the bi-parameter mixed Lipschitz spaces and characterize it via the Littlewood-Paley theory,which is between the product Lipschitz space and the inhomogeneous product Lipschitz space.We obtain a boundedness criterion of singular integral operators in mixed Journé's class on mixed Lipschitz spaces.In Charter 7,we consider the following Hausdorff operator#12 where ? can be considered as a distribution on Rn.When n?2,we obtain certain necessary and sufficient conditions for the boundedness of H? on Hp(Rn)(0<p<1).Moreover,we establish a new result of the boundedness of H? on Lp(Rn),p>1.The key idea used here is to reformulate H? as a convolution operator.