Research About The Fractional Integral Operators With Rough Kernels And Their Commutators

The boundedness of fractional integral operators in various functional s-paces is an important topic in harmonic analysis.In this dissertation,we stud-ied the boundedness of the fractional integral operators with rough kernels TΩ,αand the commutators[b,TΩ,α]generated by TΩ,α,and BMO functions in the van-ishing generalized weighted Morrey spaces and vanishing generalized Morrey spaces with variable exponents.The main structure of this paper is as follows:In Chapter 1,we introduced the background,research status of fractional integral operators,and gave relevant definitions,lemmas and the main research contents of the paper.In Chapter 2,using the properties of the weight functions A(p,q),and the pointwise estimates of TΩ,α and[b,TΩ,α],we investigated the boundedness of the fractional integral operators with rough kernels TΩ,α and the commuta-tors[b,TΩ,α]generated by TΩ,α and BMO functions in vanishing generalized weighted Morrey spaces.In Chapter 3,we studied the boundedness of the fractional integral opera-tors with rough kernels TΩ,α and the commutators[b,TΩ,α]generated by TΩ,αand BMO functions in vanishing generalized Morrey spaces with variable ex-ponents on unbounded sets by the properties of variable exponent functions and the pointwise estimates of TΩ,α and[b,TΩ,α].In Chapter 4,we summarized the conclusion of the paper.