Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms

Let L=-Δ+V be a Schrödinger operator, where Δ is the Laplacian on Rd and the nonnegative potential V belongs to the reverse Hölder class RHq for q≥d. The Riesz transform associated with the operator L=-Δ+V is denoted by R=∇(-Δ+V)-1/2 and the dual Riesz transform is denoted by R⁎=(-Δ+V)-1/2∇. In this...
Ausführliche Beschreibung

Let L=-Δ+V be a Schrödinger operator, where Δ is the Laplacian on Rd and the nonnegative potential V belongs to the reverse Hölder class RHq for q≥d. The Riesz transform associated with the operator L=-Δ+V is denoted by R=∇(-Δ+V)-1/2 and the dual Riesz transform is denoted by R⁎=(-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 class RHq for q≥d. Then we will establish the mapping properties of the operator R and its adjoint R⁎ 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 than Ap, BMO(Rd), and Lp,κ(w) corresponding to the classical Riesz transforms (V≡0). Ausführliche Beschreibung