The Boundedness Of Generalized Riesz Transforms And Fractional Integral Operators On Morrey-Herz Spaces

The boundedness of singular integral operators and their commutators on some function spaces is a very importment topic in harmonic analysis and PDE. There are lots of problems, which are related to it, such as the convergence of the Fourier series, the posedness of the solution of some PDE, and so on. Our main purpose in this paper is to obtain the boundedness of generalized fractional integral operators and Riesz transforms on homogeneous Morrey-Herz spaces, including their communators. This thesis consists three chapters.In chapter1, we introduce Herz spaces, Morrey spaces and define a class of new spaces named Morrey-Herz spaces, whose two special cases are Morrey and Herz spaces. We mainly study the generalized fractional integral operators L-β/2associated with divergence form elliptic operator and their commutator [b,Lβ/2] generated by the generalized fractional integral operators and BMO(Rn) functions. By the methods of studying ring decomposition of functions and thier corresponding truncated operators, their boundedness of the results from space MKp1,q1α,γ(Rn) to MKp2,q2α,γ(Rn) space are established.In chapter2, when the heat kernel pt(x,y) associated with second divergence form elliptic operators satisfys upperboundedness of Gaussian. On homogeneous weighted Morrey-Herz spaces, we obtain their boundedness of the generalized Riesz transform▽L-1/2associated with divergence form elliptic operator and its commutato [b,▽L-1/2] generated by generalized Riesz transform and BMO functions. Moreover, under some conditions, the boundedness for the higher order commutators RLb,m generated by generalized Riesz transform RL and CBMO functions is considered. We prove that RLb,m is bounded from MKq1,q1α,γ(Rn) spaces to MKp2,q2α1,γ(Rn) spaces.In chapter3, when the heat kernel pt(x,y) associated with divergence form elliptic operators do not satisfy upperboundedness of Gaussian, the boundedness of▽L-1/2and L-β/2is established on homogeneous Morrey-Herz spaces, when the heat kernel associated to L has L2off-diagonal estimates.