Boundedness Of The Singular Integral Operators On Weighted Morrey Spaces

Calderon and Zygmund established the singular integral theory,developed the real method of Fourier analysis on Rn,and initiated the research of modern harmonic analysis theory.Harmonic analysis includes various function space theory,weighted theory,multi-linear operator theory,harmonic analysis on homogeneous spaces and non-homogeneous spaces and other theories.In harmonic analysis,we mainly study the boundedness of the Hardy-Littlewood maximal operator,the singular integral operators and commutators,the fractional integral operators and the other operators on various function spaces.It is widely used in partial differential equation,multiple complex functions,potential the-ory,nonlinear analysis and other mathematical fields,and also in the fields of the signal processing and quantum mechanics.The Morrey spaces were introduced in order to investigate the local behavior of solutions to second order elliptic partial differential equations.Chiarenza and Frasca showed the boundedness of the Hardy-Littlewood maximal operator,the singular integral operators and the fractional integral operators on the Morrey spaces.Komori and Shirai introduced the weighted Morrey spaces and proved that the classic operators in harmonic analysis are bounded on the weighted Morrey spaces.In recent years,the boundedness of various singular integral operators on weighted Morrey spaces has become one of the hot spots in the field of harmonic analysis.At present,many important achievements have been made,but there are still many problems to be studied.In this paper,we obtain the endpoint estimates for the commutators of the singular integral operators with the BMO functions and the associated maximal operators,the commutators of the fractional integral operators with the BMO functions and the asso-ciated maximal operators on the weighted Morrey Spaces and the generalized weighted Morrey spaces.We also get the boundedness of the Hardy-Littlewood maximal operator,the multilinear Hardy-Littlewood maximal operator,the multilinear singular integral op-erators and the commutators of multilinear singular integral operators with the BMO functions on the weighted Morrey spaces.This paper consists of four chapters.In the introduction,the background,current research status of the Morrey spaces and the main operators of this paper are introduced.In chapter 1,we obtain the weighted endpoint estimates for the commutators of the singular integral operators with BMO functions and associated maximal operators on weighted Morrey Spaces.We also obtain the similar results for the commutators of the fractional integral operators with BMO functions and associated maximal operators.In chapter 2,we investigate the boundedness of the Hardy-Littlewood maximal oper-ator and the multilinear maximal operator on the weighted Morrey spaces and the weight-ed weak Morrey space.As a corollary,we get the boundedness of Hardy-Littlewood max-imal operator on the weighted Morrey spaces defined by Samko and the different proof for the boundedness results obtained by Komori and Shirai for the Hardy-Littlewood maximal operator on the weighted Morrey spaces.In chapter 3,we discuss the boundedness of the multilinear Calderon-Zygmund op-erators and the commutators of multilinear Calderon-Zygmund operators with BMO functions on the weighted Morrey spaces and the weighted weak Morrey space.We al-so get the boundedness of these operators on the Samko type weighted Morrey spaces Lp,k(w,dx).In chapter 4,we obtain the endpoint boundedness for the commutators of the singular integral operators with BMO functions and associated maximal operators on generalized weighted Morrey spaces.In the meantime,we define a kind of generalized weighted weak Morrey spaces and obtain that the commutators of the fractional integral operators with BMO functions and the associated maximal operators map the generalized weighted Morrey spaces to the generalized weighted weak Morrey spaces.