Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms. (24th April 2019) - Wang, Hua
Abstract : LetL = - Δ + V be a Schrödinger operator, whereΔ is the Laplacian onR d and the nonnegative potentialV belongs to the reverse Hölder classR H q forq ≥ d . The Riesz transform associated with the operatorL = - Δ + V is denoted byR = ∇ ( - Δ + V ) - 1 / 2 and the dual Riesz transform is denoted byR ⁎ = ( - Δ + V ) - 1 / 2 ∇ . In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder classR H q forq ≥ d . Then we will establish the mapping properties of the operatorR and its adjointR ⁎ on these new spaces. Furthermore, the weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators[ b, R ] and[ b, R ⁎ ] are also obtained. The classes of weights, classes of symbol functions, and weighted Morrey spaces discussed in this paper are larger thanA p, B M O ( R d ), andL p, κ ( w ) corresponding to the classical Riesz transforms (V ≡ 0 ).