The Estimation Of The Commutators Of Hardy Operators And CMO Functions On Some Hardy Type Spaces

This thesis is to discuss the efect of commutators Hb, H bposed by the Hardyoperator and CMO functions on Hardy type spaces Hb;p,q,satomand LHb;p,q,satom.The thesis is divided into four chapters. The frst issue of concern is thebackground knowledge and the description of the work for this paper in qualitativemethod. As for the background knowledge, it includes classic Hardy operator andHardy inequality. The second chapter presents preparation knowledge which isclosely related to this research. It is noted that the sign and lemma for the mainwork are brought out in chapter1and2. The third chapter deals with the specifcworks or key points in this thesis. Firstly, it provides the specifc defnition tothe object involves atom (b; p, q, s), L atom (b; p, q, s), Hb;p,q,sq,satomand LHb;p,atom.Through studying the impact of commutators Hb, H bon atom (b; p, q, s) andL-atom (b; p, q, s), it is revealed the fact that when a(x) is a L atom (b; p, q, s),then C1q,bHba(x) will be an atom (p, q, s) and when a(x) is a atom (b; p, q, s),(C q,b)1H ba(x) will be an atom (p, q, s1). furthermore, we yet obtain theefect and estimation of Commutators Hb;p,q,sb, H,sbon Hb;p,qatomand LHatom.The last chapter is the summary and prospects of the whole research, whichpoints out the signifcance, defciency, and the future development direction.