H - L Great Function In The Some Space In The Estimate

The paper is a survey on the estimates of the Hardy-Littlewood maximal function in some functional spaces. It contains four chapters. The outline of the paper is arranged as follows:In chapter1, we introduce three different H-L maximal functions, and then give the known results about their strong and weak norms. Moreover, some open problems are also showed.In chapter2, it is devoted to study the strong and weak estimates of H-L maximal function in some spaces. At the same time, we generalize some results of the former-chapter to these spaces. And we also find that these estimates play an important part in studying other operators, such as fractional operators.In chapter3, weight function ω is introduced and the estimates of H-L maximal function are investigated in some weighted spaces by charactering ω.In chapter4, we present the estimates of H-L maximal function in generalized Lp spaces and Morrey spaces. The results in this chapter can be applied to elasticity theory, fluid mechanics and differential equations.