Weighted Morrey Spaces Related to Schrödinger Operators with Nonnegative Potentials and Fractional Integrals

Abstract

Let L = −∆ + V be a Schrödinger operator on R d , d ≥ 3, where ∆ is the Laplacian operator on R d and the nonnegative potential V belongs to the reverse Hölder class RH s for s ≥ d/2. For given 0 < α < d, the fractional integrals associated to the Schrödinger operator L is defined byIn this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RH s for s ≥ d/2. Then we will establish the boundedness properties of the fractional integrals I α on these new spaces. Furthermore, weighted strong-type estimate for the corresponding commutator [b, I α ] in the framework of Morrey spaces 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 A p,q , BMO(R d ) and L p,κ (µ, ν) corresponding to the classical case (that is V ≡ 0).