Some Estimates For Multilinear Fractional Integral Operators And Littlewood-Paley Operators

Multilinear operator theory is a natural generalization of the linear operator theory.S-ince its widely application in partial differential equations,the multilinear operator theory has been developed rapidly in the field of harmonic analysis and has received many schol-ar’s attention.In this paper,we consider mainly some estimates for multilinear fractional integral operators and multilinear Littlewood-Paley operators.In chapter one,the research background,current situation of domestic and foreign research for multilinear operators are reviewed.Furthermore,we simply introduce the main structure arrangement in this paper.In chapter two,T.Iida et al introduced new Morrey type spaces MPp0 and proved a new Morrey type norm was strictly smaller than m-fold product of the Morrey spaces in 2012.Therefore,we study the endpoint estimates for multilinear fractional integral opera-tors Iα,m on Morrey type spaces,namely operators Iα,m are bounded from MPp0 to BMO and from MPp0 to Lipα-n/p0,which are more accurate than previous results.In chapter three,we establish the necessary and sufficient condition such that the modified multilinear fractional integral operators Tα are bounded from εβ1,p1×…×εβm,pm to εβ,p spaces and from Bs1×Πi=2nεβi,pi to Bs,where Bs is homogeneous Sobolev spaces.The results are also new for the case m=1.In chapter four,we discuss the estimates for multilinear Littlewood-Paley opera-tors with non-convolution type kernels on Campanato spaces,namely[g(f)]2 is bounded from εβ1,p1×…×εβm,pm to ε2β,p/2.Moreover,we also establish similar results for the Lusin area function S and the Littlewood-Paley gλ*-function.