Boundedness And Compactness For Multilinear Singular Integral Operators

Since A.P.Calderón combined the commutator with the multilinear operator the-ory,the related research on the multilinear singular integral operators and their commu-tators has attracted the attention of many scholars.At present,their properties in various function spaces have achieved fruitful results,but there are still many problems to be solved.The whole thesis is divided into four chapters.In Chapter 1,we sort out the de-velopment process and existing conclusions of the multilinear singular integral oper-ators and their commutators,find out some problems that remain to be solved and have academic value,and determine the topic to be studied in this thesis: the bound-edness for the iterated commutators of multilinear Calderón-Zygmund operators with RVMO(?)functions on the product Morrey spaces with non-doubling measures and the compactness for the iterated commutators of multilinear ?-type generalized fractional integral operators with RBMO(?)functions on the product Lebesgue spaces under non-homogeneous metric measures.In Chapter 2,the compactness criterion of the multilinear integral operators on the product Lebesgue spaces is given,firstly.Then by the method of function approxi-mation,it is proved that,under the non-doubling measures,the iterated commutators of multilinear Calderón-Zygmund operators with C?c(Rd)functions are compact on the product Morrey spaces.Thus,the compactness for the iterated commutators gener-ated by the multilinear Calderón-Zygmund operators and RVMO(?)functions on the product Morrey spaces,under the non-doubling measures,is given.Finally the above conclusion is generalized to the generalized product Morrey spaces.In Chapter 3,we consider the cases with more general measures and the non-homogeneous metric spaces.We first prove the boundedness of the multilinear ?-type generalized fractional integral operators in the product Lebesgue spaces.Then,by es-timating the sharp maximal function,the boundedness for the iterated commutators of the multilinear ?-type generalized fractional integral operators in the product Lebesgue spaces is given.In Chapter 4,we summarize the whole thesis.