The Boundedness Of Multilinear Singular Integrals On Weighted Lorentz Spaces

In this paper, we mainly discuss the boundedness of multilinear maximal functions andmultilinear singular integrals on the weighted Lorentz spaces. Firstly, we obtain the bounded-ness of singular integrals and its commutators on the weighted Lorentz spaces. Secondly, westudy the boundedness of the multilinear maximal functions, the multilinear singular integralsand its commutators on the weighted Lorentz spaces. Finally, we discuss the three weight weaktype inequalities for the multilinear fractional maximal functions.In section1, we introduce some relevant preliminaries and give some conclusions whichwe needed in this paper briefly.In section2, Firstly, we obtain the Fefferman-Stein inequalities associated the maximalfunction with the sharp maximal function on the weighted Lorentz spaces. By applying theFefferman-Stein inequalities, the sharp estimates of the singular integrals and the commuta-tors, and the boundedness of maximal functions on the weighted Lorentz spaces, we obtain theboundedness of the singular integrals and the commutators on these spaces.In section3, we obtain the boundedness of the multilinear maximal functions, the mul-tilinear singular integrals and the commutators on the weighted Lorentz spaces. Firstly, weintroduce the definitions of the weight classes A(?) and obtain which are the necessary con-ditions of weak type inequalities of the multilinear maximal functions. Secondly, we discussthe sufficient conditions for the weak type inequalities of the multilinear maximal functions.we also introduce the weak boundedness of the multilinear maximal functions (?). Final-ly, we obtain the boundedness of the multilinear singular integrals and the commutators on theweighted Lorentz spaces by using their sharp estimates.In section4, we study the necessary and sufficient conditions of the three weight weaktype inequalities for the multilinear fractional maximal functions. Finally, we introduce anoth-er multilinear maximal function (?)(?) which under our assumptions majorizes the functionM_γ (?).