Boundedness And Compactness For Commutators Of Some Fractional Integral Operators On Morrey Spaces

The Morrey spaces were originally proposed by American mathematician C.B.Mor-rey in 1938 to study the continuity of solutions of partial differential equations in the class of measurable functions.At the beginning of the 21st century,Y.Sawano and H.Tanaka gave the definition of Morrey space under the condition of non-doubling measures.With the progress of weighted theory,Y.Komori and S.Shirai defined the weighted Morrey spaces.In recent years,more and more scholars have started to study the boundedness and compactness of various operators and commutators on different types of Morrey spaces,which has become a hot topic in the field of harmonic analy-sis.Many important results have been obtained,but there are still many problems that need to be further explored.This paper is divided into four chapters.In Chapter 1,we systematically review the historical development and existing research results of bilinear fractional integral operators and their commutators on Mor-rey spaces,and identify some academically valuable but unresolved problems,which mainly include:the compactness of iterated commutators generated by generalized bi-linear fractional integral and RCMO（μ）functions on non-doubling measure Morrey spaces,and the boundedness and compactness of bilinear fractional maximal linear com-mutators and iterated commutators on weighted Morrey spaces.In Chapter 2,we first give the boundedness of the generalized bilinear fractional integral operatorK_αon Lebesgue space and Morrey space under non-doubling mea-sure conditions,and then obtain the boundedness of the generalized bilinear fractional iterated commutators on non-doubling measure Morrey spaces,and then prove its com-pactness using function approximation.In Chapter 3,we consider the weighted Morrey spaces,and we first prove the boundedness of the bilinear fractional maximal linear commutatorsM_α,bⁱand the bilin-ear fractional maximal iterated commutators_Mα,b,and then use the Fréchet-Kolmogorov theorem and function decomposition to prove their compactness on weighted Morrey spaces.In Chapter 4,we summarize the previous research and give an outlook on the future directions that can be studied.