Several Results About The Boundedness Of The Operator

By the theory of atomic and the molecular decompositions of Herz type Hardy spaces and using the Morrey-Herz space with the characterization of Herz space, the boundedness of a kind of operators is proved. The results enrich theory of Herz space, we put this thesis into the following three parts:The first part, by using the atomic decompositions of the weighted Herz type Hardy space, the boundedness of the multilinear commutators of Bochner-Riesz operators on the weighted Herz type Hardy space is obtained.The second part, by using the decompositions of the Herz type Hardy space, the fractional integrals operator with variable kernel are discussed. When some conditions are given about the kernel,the boundedness of such kind of operators in the Herz type Hardy spaces is proved, and the boundedness of the commutators generated by the fractional integral with Lipschitz functions is showed.At the last part, the boundedness for the commutators of the fractional Hardy operators with RBMO(?) function on Morrey-Herz space with non-doubling measure is discussed. By using the Morrey-Herz spaces with the characterization of Herz spaces and the properties of the RBMO(?) functions enjoying with the classical BMO. We show that the commutators of fractional Hardy operators is bounded on Morrey-Herz spaces with non-doubling measure.