Boundedness Of Marcinkiewicz Integrals And Commutators On Variable Exponent Spaces

In this dissertation, we shall deal with the the boundedness of a class of Marcinkiewicz integrals operators and its commutators on some function spaces. The main results are as follows:In the chapter 1, the boundedness of a class of Marcinkiewicz integrals oper-ators and its commutators is considered. By using the boundedness properties on Lebesgue spaces with variable exponent, the boundedness results of Marcinkiewicz integrals μ, Lusin area integrals μs, and Littlewood-Paley gλ* and their commutators are established on Morrey spaces with variable exponent.In the chapter 2, the boundedness of a class of Marcinkiewicz integrals operators μ and its commutators [b,μ] is considered. By using the boundedness of μ and [b,μ] on Lebesgue spaces with variable exponent, we prove that Marcinkiewicz integrals operators and its commutators are bounded on Herz-Morrey spaces with variable exponent.In the chapter 3, the boundedness of a class of singular integral operators T and its commutators [b, T] is considered. By using the boundedness of T and [b, T] on Lebesgue spaces with variable exponent, we prove that singular integral operators and its commutators are bounded on the homogeneous and non-homogeneous Herz spaces with variable exponent.In the chapter 4, the boundedness of a class of multilinear singular integral-s of Calderon-Zygmund type T and is considered. By using the boundedness of T on Lebesgue spaces with variable exponent, we prove that multilinear Calderon-Zygmund singular integrals operators are bounded on Herz-Morrey spaces with vari-able exponent.