Classes of weights and second order Riesz transforms associated to Schrödinger operators

Abstract

We consider the Schrödinger operator −Δ + V on ℝⁿ with n ≥ 3 and V a member of the reverse Hölder class ℬ_s for some s > n/2. We obtain the boundedness of the second order Riesz transform ∇² (−Δ + V)⁻¹ on the weighted spaces L^p(w) where w belongs to a class of weights related to V. To prove this, we develop a good-λ inequality adapted to this setting along with some new heat kernel estimates.