Boundedness Of Singular Integral Commutators Acting On Hardy Spaces

Harmonic Analysis is a new branch of fundamental mathematics,which primarily studies some function spaces and operators and plays an important role in partial differ-ential equation.For any researcher who work about analysis and PDEs,it is an indis-pensible tool.Recently,Calder(?)n-Zygmund singular operator and Monge-Amp(?)re sin-gular operator regarded as classical operators in Harmonic Analysis are attracting many researchers' eyes whose many essential properties are obtained.the authors study the boundedness of mixed multilinear commutators[(?),[(?),T]]generated by the BMO func-tions Bi(i = 1,…,m)and Lipschitz functions bj(j = 1,…,l)on the Lebesgue space and Hadry space.Then the authors obtain that this multilinear commutators is bounded from Lp(Rn)to Lq(Rn)and Hb,B1(Rn)to Ln/n-?(Rn).Furthermore,the authors also prove that when m = 1,this multilinear commutators are bounded from H1(Rn)to Ln/n-?,?(Rn).In addition,let H be Monge-Amp(?)re singular integral operator,b ? Lip?F and 1/q = 1/p-?,we also discus the behaviors of commutators of Monge-Amp(?)re operator act on Lebesgue spaces and Hardy spaces in this paper.It is proved that the commutator[b,H]is bound-ed from Lp(Rn,d?)to Lq(Rn,d?)for 1<p<1/? and from HFp(Rn)to Lq(Rn,d?)for 1/(1+?)<p ? 1.For the extreme case p = 1/(1 +?),the weak estimate is given.There are four chapters in this paper:In chapter 1,we will introduce about the background,current research situation and the frame of the paper.In chapter 2,we discuss bounded-ness of the mixed multi-commutator[(?),[(?),T]]act Lebesgue spaces and Hardy spaces.In chapter 3,we study boundedness of the commutator[b,H]act Lebesgue spaces and Hardy spaces.