Weighted Morrey Spaces Related to Schrödinger Operators with Nonnegative Potentials and Fractional Integrals. (16th March 2020) - Wang, Hua
Abstract : Let ℒ = − Δ + V be a Schrödinger operator on ℝ d, d ≥ 3, where Δ is the Laplacian operator on ℝ d, and the nonnegative potential V belongs to the reverse Hölder class RH s with s ≥ d / 2 . For given 0 < α < d, the fractional integrals associated with the Schrödinger operator ℒ is defined by ℐ α = ℒ − α / 2 . Suppose that b is a locally integrable function on ℝ d and the commutator generated by b and ℐ α is defined by b . ℐ α f x = b x ⋅ ℐ α f x − ℐ α b f x . In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class RH s with 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 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 A p, q, BMO ℝ d, and L p, κ μ, ν corresponding to the classical case (that is V ≡ 0 ).