Fractional Carleson Measure Associated With Hermite Operator

One of the important research objects of Modern Real Analysis is function spaces,especially based on the course of Real Variable Function Theory and Functional Analysis,focusing on the study of the various function spaces composed of real variable functions,and the equivalent characterizations of these spaces and the boundedness of singular integral operators on these function spaces.The major function spaces include Lebesgue space,continuous function space,Hardy space,bounded mean oscillation function space BMO,etc.Function space theory forms an important part of the Harmonic Analysis,and they provide a very suitable framework for the study of Modern Analysis.In recent years,the study of function spaces associated with the Hermite operators has inspired numerous interests.Teacher Huang Jizheng used the Lusin area integral operator to describe Hardy space.Later,Teacher Huang Jizheng and Sister Wang Yaqiong utilized the Carleson measure related to Hermite operator to characterize the bounded mean oscillation function space BMO.The purpose of this thesis is to further study and expand this topic taken into account their research.One of the main purposes of studying the function spaces is to give the equivalent characterizations of them.For example,the heat maximal functions and square functions were used to characterize the classical Hardy spaces,and the Carleson measure was used to characterize the classical BMO spaces or MorreyCampanato spaces and so on.In this thesis,we first define the fractional Carleson measure associated with the Hermite operator,then we use fractional Carleson measure to give equivalent characterizations of predual spaces of the BMO spaces and Hardy spaces associated with Hermite operator.In this thesis,we first introduce the fractional semigroups,and utilize the subordinative formula to obtain the corresponding estimates of the kernels;Secondly,the integral kernel of Poisson semigroup is accurately estimated;Then we give the Carleson measure characterization of the dual space Campanato space of Hardy space associated with the Hermite operator;Finally,the equivalent characterization of the predual spaces of the Hardy spaces associated with the Hermite operator are given.