Boundedness Of Multilinear Singular Integral Operators And Multilinear Commutators

In this dissertation, the author mainly consider the following three kinds of questions: the boundedness of multilinear operators on generalized Morrey spaces over the quasi-metric space of nonhomogeneous type; the estimates of some multi-linear operators with rough kernel on weighted Morrey spaces; the boundedness of multilinear commutators of generalized fractional integral operators on weighted Morrey spaces.The thesis is composed of five chapters.In chapter one, we mainly introduce the background of multilinear operators and their commutators, the present situation of the research and our work of this paper.In chapter two, on the basis of generalized Morrey spaces with nondoubling measures introduced by Sawano [37], we generalized the Euclidean space Rn to abstract space X. and define a quasi-metric as well as generalized Morrey space Lp,φ(X,k,μ) on it. We obtain the boundedness of multilinear fractional integral operators Tα,m, the Calderon-Zygmund operators T and the multi-sublinear max-imal operators Mκ on generalized Morrey spaces over the quasi-metric space of nonhomogeneous type. The results of the boundedness of these operators on Mor-rey spaces with nondoubling measures have been generalized.In chapter three, we study the multilinear fractional operators with rough ker-nel TΩ,αA and its corresponding maximal operators MΩ,α,A where Ω∈Ls(Sn-1)(s>1) is homogeneous of degree zero, Sn-1is the unit sphere of Rn. Let α=0, then the above operators are the multilinear singular integral operators with rough ker-nel TΩA and its corresponding maximal operators MΩA. When DγA∈Aγ(Rn)(β-th order Lipschitz space)(|γ|=m-1), the boundedness of the above operators on weighted Morrey spaces Lp,κ(u,v) are obtained. When Dγ A∈BMO(Rn)(bounded mean oscillation space)(|γ|=m—1), TΩ,α,A MΩ,α,A are bounded on Lp,κ(u,v); while for TΩ,A MΩ,A we only consider the case m=1, m=2, and we prove their boundedness on weighted Morrey spaces with one weight Lp,κ(w).In chapter four, we consider the multilinear commutators Lb/-α/2generated by the generalized fractional integral and a vector function b=(61,…,bm), where bi, i=1,…,m, are weighted BMO functions, e-tL is an analytic semigroup with a kernel Pt(x, y) generated by a linear operator L on L2(Kn), the kernel pt(x,y) satisfies a Gaussian upper bound, that is, for x, y∈Rn ard all t>0. The boundedness of Lb-α/2on weighted Morrey space is obtained. Our work in this chapter generalize commutator [b, L-α/2] to multi-[b, L-α/2] linear commutator L-α/2and improve the corresponding results of commutatorIn chapter five, we conclude our work in this thesis. Moreover, we point out more questions will be considered on the basis of our research. Whether can we and how to improve our results.