On The Boundedness Of Maximal Operators

Maximal operators is an important tool for harmonic analysis research.It has a wide range of applications in the proof of Lebesgue differential theorem and pointwise estimation.Although it has been extensively studied for decades,there are still some unsolved problems with maximal operators.Therefore,understanding the boundedness of maximal operators in different function spaces is still a hot research topic.This article is divided into three parts.The first part introduces the concepts of classical Hardy-Littlewood maximal operators and fractional maximal operators,definitions of some function spaces and related properties.The second part sorts out the boundedness of this two types of maximal operators in Sobolev spaces and Morrey spaces.The third part proves the boundedness of this two types of maximal operators in Sobolev-Morrey space W1,p,γ(Rn):1、Classical H-L maximal operators is bounded from W1,p,γ(Rn)to W1,p,γ(Rn),1<p<∞,0 ≤γ≤n and|DiMf|≤MDif,(?)x∈(Rn)i= 1,2,…n.2、Fractional maximal operators is bounded from W1,p,γ(Rn)to W1,q,t,1<p<∞,0≤β≤<n/p,0≤γ≤n,1<t≤q<∞,1/q=1/p-α/n,t/q=γ/p,and|DiMβf|≤MβDif (?)x∈(Rn)i= 1,2,…n.The fourth part is a summary of this article,and puts forward some questions that can be studied in the future.