Study For The Boundedness Of A Class Of Multilinear Singual Integral

In this dissertation, the author mainly consider two questions: On one hand, thisdissertation is devoted to introduction of Calder′on-Zygmund operator and give moregeneral multi-commutators with Calder′on-Zygmund kernel, on the other hand, which isdevoted to introduction of multilinear fractional integral operators. The work of thispaper is as follows:In chapter 1. Introducing the background of multilinear singular integral operators,the present situation of the research and the relevant prepare knowledge.In chapter 2. we obtain that the boundedness from Bp(w) spaces to CMO(w) spacesfor the multi-commutators generalized by Calder′on-Zygmund operators and the (m + 1)-th remainder of Taylor series of the functions whose m-th derivatives belong to BMOspaces. As corollary, the Lp-boundedness for the multi-commutators will be obtained.In chapter 3. firstly, we prove that the multilinear fractional integral operator and therelevant maximal operator with rough kernel are both bounded operators from Lp(wp)to Lq(wq) with w∈A(p,q) where Dγ(Ai)∈∧˙β(Rn) or Dγ(Ai)∈Lri(Rn), from L1 toLn/(n-α),∞with power weight and from Lp(1 < p <∞) to Lq where Dγ(Ai)∈BMO(Rn),from Lp to L n-nα,∞with power weight and from Lp to Ln/(n-α),∞where Dγ(Ai)∈Lri(Rn),i = 1,2,···,k; secondly, we consider that the boundedness of the multilinear fractionalintegral operator with rough kernel from H1(Rn) to L n-nα,∞(Rn) for related to the mi-thremainder of Taylor series of Ai at x about y for i = 1,2,···,l; thirdly, we get that themultilinear fractional integral operator and the relevant maximal operator with variablekernel are both bounded operators from Lp(1 < p < n/α) to Lq, from H1,∞to L n-αn-kβ,∞where Dγ(Ai)∈∧˙β(Rn) and from Lp(1 < p < n/α) to Lq, from H1,∞to L n-nα,∞whereDγ(Ai)∈BMO(Rn), i = 1,2,···,k.