Boundedness Of The C-Z Singular Integral Operator And Commutator On Herz Type Spaces With Variable Exponents

To solve the inconsistency caused by non-linear problems in natural science and engineering technology,the theory of function spaces with variable exponents generalized by classical function spaces has attracted many scholars' attention and research.The properties of operators on function spaces with variable exponents have been one of the key issues in harmonic analysis,which is closely related to partial differential equations with non-standard growth conditions,electrorheological fluids and image restoration.In this paper,we mainly discuss the boundedness of C-Z singular integral operator and their commutator on Herz type spaces with variable exponents.The details are as follows:(1)The definitions of C-Z singular integral operator,commutator generated by BMO function and Herz space with variable exponents are given in the paper.Then we discuss the boundedness of C-Z singular integral operator and its commutator on Herz space with variable exponents,and prove the results with relevant lemma about Ls-Dini condition,spherical area integral formula,the boundedness of C-Z singular integral operator or commutator on Lq(·)space and other lemmas.(2)The original definition and equivalent norm definition of Herz space with variable exponents are introduced.Then we apply the equivalent definition of Herz space with variable exponents and atomic decomposition of Herz-type Hardy space with variable exponents to verify the boundedness of C-Z singular integral operator and conunutator from Herz-type Hardy spaces with variable exponents to Herz spaces with variable exponents.