Weighted Morrey Spaces Related to Schr?dinger Operators with Nonnegative Potentials and Fractional Integrals

摘要

Let be a Schr?dinger operator on , , where is the Laplacian operator on , and the nonnegative potential V belongs to the reverse H?lder class with . For given , the fractional integrals associated with the Schr?dinger operator is defined by . Suppose that b is a locally integrable function on and the commutator generated by b and is defined by . In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse H?lder class with . Then, we will establish the boundedness properties of the fractional integrals on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator in the framework of Morrey space is also obtained. The classes of weights, the classes of symbol functions, as well as weighted Morrey spaces discussed in this paper are larger than , , and corresponding to the classical case (that is ).