Marcinkiewicz Integral High-order Commutator Related To Schrödinger Operator

In this thesis,firstly,the definitions of the Marcinkiewicz integral operator,the high order commutator of Marcinkiewicz integral associated with Schr?odinger operator,the generalized Morrey space related to nonnegative potential,the spaces with variable exponents are introduced.Some basic properties of Marcinkiewicz integrals and the space with variable exponents are stated.Then,by using the classical inequalities and properties of the Marcinkiewicz integrals,the boundedness of the high order commutator generated by Marcinkiewicz integrals with rough kernel associated with Schr?odinger operators and Campanato class on Lebesgue space is obtained.We proved that the high order commutator is also bounded on the generalized Morrey space related to nonnegative potential.Finally,using the properties of Lebesgue space with variable exponents and the boundedness of high order commutator on Lebesgue space with variable exponents,based on the definition of the Herz-Morrey space with variable exponents,the boundedness of high order commutator generated by Marcinkiewicz integrals with Lipschitz kernel associated with Schr?odinger operators and BMO functions on Herz-Morrey with variable exponents is established.