The Boundedness For Schr(?)dinger Type Operators Related To Certain Nonnegative Potentials In Generalized Morrey Spaces

In this thesis,we mainly study the boundedness of some relevant Schr(?)dinger type operators with a potential function satisfying the so-called reverse H(?)lder class Bn,and general non-divergence Schr(?)dinger operator with discontinuous coefficients in gener-alized Morrey spaces associated with potential,respectively.The first one is to study the boundedness for commutators generated by generalized Campanato functions and the Riesz transforms of standard Schr(?)dinger operators in generalized Morrey spaces asso-ciated with potential.The second one is to show the boundedness of non-divergence Schr(?)dinger type operators with discontinuous coefficients in generalized Morrey spaces associated with potential.More precisely,the contents are shown as follows.Firstly,we consider the following Schr(?)dinger operators with potentials:L=-?+V(x),where ? is the Laplacian on on Rn(n>3),and V(x)is a nonnegative potential belonging to the class Bn satisfying reverse H(?)lder inequality.Let b belong to a generalized Campanato space ?v?(p),and T be one of the following Schr(?)dinger type operators ?(-?+V)-1?,?(-?+V)-1/2,(-?+V)-1/2?.The purpose of this first part is to study the boundedness of the commutators[b,T]generated by the functions b and Schr(?)dinger type operators T in generalized Morrey spaces Mp,? ?,V and vanishing generalized Morrey spaces VMp,? ?,V associated with potential,respectively.Specifically speaking,if the function b belongs to?v?(?)and the weight pair(?1,?2)satisfies some conditions,then we conclude that[b,T]is bounded from the generalized Morrey spaces Mp,?1 ?,V to Mq,?2 ?,V for any 1<p<? and 1/p-1/q=v/n.This is bounded from M1,?1 ?,V to WMn/n-v,?2 ?,V(the weak generalized Morrey spaces)for the borderline case p=1.Furthermore,we prove that[b,T]is bounded from the vanishing generalized Morrey space VMp,?1 ?,V to VMq,?2 ?,V for any 1<p<?,and it is proved to be bounded from VM1,?1 ?,V to VWMn/n-v,?2 ?,V(the vanishing weak generalized Morrey spaces)for the borderline case p=1.We devote the second part to consider the following non-divergence Schr(?)dinger operators with discontinuous coefficient L1=-ai,j(x)Di,j+V(x),where ai,j(x)? BMO?(p)(the space of generalized bounded mean oscillation),and the nonnegative potential V(x)belongs to Bn/2*(the expanded class of reverse H(?)lder inequal-ity).In order to show the boundedness of the Schr(?)dinger equation L1u=f,we are to find the sufficient conditions on the weight pair(cp1,?2)which ensure the boundedness of non-divergence form Schr(?)dinger operators ?2L1-1 in generalized Morrey spaces Mp,? ?,V associated with potential,provided the semi-norm of the discontinuous coefficient ai,j(x)is sufficiently small in the generalized bounded mean oscillation space BMO?(p).That is to say,if 1<p<? and V2L1-1?this operator is bounded from M_p,?^?,V to M_p,?2^?,V.