Boundedness Of Commutators Of Some Integral Operators

This article is divided into four chapters,we mainly discuss the boundedness of commutators of ?-type Calderon-Zygmund operator,commutators of strongly singular integral operator and commutators of maximal function.In Chapter 1,we mainly introduce the research background,significance and progress of commutators of ?-type Calderon-Zygmund operator,commutators of strongly singular integral operator,commutators of maximal function,and put forward the questions to be studied.In Chapter 2,we study the commutators generated by the ?-type CalderónZygmund operator and the locally integrable function b,and discuss two problems.The first problem is proved that the commutator is bounded from Lebesgue space Lp(Rn)to Campanato space Cp,?(Rn)when b???(Rn)(?=?+n/p);The second problem is that the commutator is bounded from Morrey space to Campanato space when b?Cp,?(Rn).In Chapter 3,we study the commutators generated by strongly singular integral operators and locally integrable function b.Two results are obtained.First,we prove the commutator is bounded from Lebesgue space Lp(Rn)to Campanato space Cp,?(Rn)when b belongs to Lipschitz function class;Second,we prove the boundedness of commutators from Morrey spaces to Campanato spaces when b belongs to Campanato spaces.In Chapter 4,the maximal commutator Mb and commutator[b,M]generated by maximal function M and weighted Lipschitz function b are discussed,and proved that b?Lip?,? is a necessary and sufficient condition for Mb to be bounded from Lp(?)to Lq(?1-q),and it is also proved that the necessary and sufficient condition for[b,M]to be bounded from Lp(?)to Lq(?1-q)is b?Lip?,? and b?0,a.e.x?Rn.