Marcinkiewicz Integrals And Their Commutators On Mixed Norm Spaces

The disseration mainly studies the weighted boundedness of Marcinkiewicz integral on mixed norm Morrey space,the weighted boundedness of Marcinkiewicz integral associated with Schrodinger operator on mixed norm Morrey space and weighted boundness estimation of Marcinkiewicz integral with variable kernel associated with Schrodinger operator on generalized Morrey Spaces with vanishing variable exponent.The main results are as follows.Firstly,the use of A（p,q）weight and A_p weight estimates,combined with the tools of Holder inequality and Minkowski inequality and the methods of function decomposition,such as,by using of the weighted boundness of Marcinkiewicz integral,Riesz potential and their commutators on L^p Spaces.The weighted boundness of Riesz potential and Marcinkiewicz integral on mixed norm Morrey Spaces are proved and the weighted boundness of the BMO commutators of Riesz potentials and Marcinkiewicz integrals are obtained.Secondly,by using the estimation of A_p weight,combined with the tools of Holder inequality and Minkowski inequality and the method of function decomposition,by using the weighted estimation of Marcinkiewicz integral and its commutator related to Schrodinger operator on L^p space.The weighted boundedness of the Marcinkiewicz integral associated with Schrodinger operator on mixed norm Morrey Spaces is proved and the weighted boundedness of the BMO commutator of the Marcinkiewicz integral associated with Schrodinger operator is given.Finally,by using the estimation of A_<sup>p（·）weight,combined with some tools such as Holder inequality and Minkowski inequality and the method of function decomposition.By means of the weighted estimation of variable kernel Marcinkiewicz integral associated with Schrodinger operator and its commutator on L^<sup>p（·）space,the boundness of variable kernel Marcinkiewicz integral and its commutator related to Schrodinger operator in the generalized weighted Morrey space with variable exponent are proved.The weighted boundedness of variable kernel Marcinkiewicz integral associated with Schrodinger operator and its commutator on vanishing generalized Morrey space with variable exponent are given.