The Boundness Of Area Functions Associated With Differential Operators On Product Domain

The area integral is one of the important topics in harmonic analysis.It can be used to characterize real Hardy spaces and study the regularity of the solutions of elliptic equations on domains,etc.In recent years,the harmonic analysis associated with differential operators has become one of the important contents of harmonic analysis.In this thesis,we will investigate the boundedness of the area integral as-sociated with the differential operators on the product spaces.As an application,when L is a second order differential operator in divergence type,high order elliptic differential operator with constant coefficients and high order elliptic differential op-erator in divergence type,we obtain the boundedness of the area integral connnected with the operator L on the product spaces.The second chapter mainly introduces the basic ssymbols and the Calderon-Zygmund decomposition of vector valued functions.In the third chapter,we intro-duce the off-diagonal estimates,H? functional calculus and some basic assump-tions about operators.In the fourth chapter,the definitions of the area integral asso-ciated with the operators in the case of vector value and the area integral connected with the operators on the product spaces are given.We also give their boundedness results.The fifth chapter is the proofs of the main theorems given in the fourth chapter.In the sixth chapter,we apply the main theorems in the fourth chapter to three kinds of differential operators:second order differential operators in diver-gence type,high order elliptic differential operators with constant coefficients and high order elliptic differential operators in divergence type.