Boundedness Of Integral Operators On Local Morrey Space

The study of boundedness of operators in function spaces is one of the hot topics in harmonic analysis,the methods and techniques developed from which have been widely used in the field of partial differential equations.In 1938,Morrey introduced Morrey spaces,and the boundedness of various operators on Morrey spaces was successively obtained.With the development of weighted theory,Komori and Shirai defined the weighted Morrey spaces,and obtained the boundedness of singular integral operators and fractional integral operators on weighted Morrey spaces.Inspired by the above,we study the boundedness of β-local singular integral operators and β-local fractional integral operators on Morrey spaces with local weights.First,two types of local weights are introduced,and some simple properties of local Ap,q weights are obtained.Secondly,the Morrey spaces with local Ap weights are introduced,and the boundedness of β-local singular integral operators on local Morrey spaces are obtained by means of function decomposition and local Whitney covering lemma.Finally,the Morrey spaces with local two-weight are defined,and the boundedness of β-local fractional integral operators and the associated fractional maximal operator on local Morrey spaces are also obtained by means of function decomposition and local Whitney covering lemma.The combination of Morrey spaces with local weights and the boundedness of singular integral operators enrich the theory of Morrey spaces.