Riesz transforms, fractional power and functional calculus of Schrödinger operators on weighted [formula omitted]-spaces

Abstract

We consider the Schrödinger operator A=−Δ+V+−V− on Lp(RN,wdx) where N≥3, and w is a weight in some Muckenhoupt class. We study the boundedness of Riesz transform type operators ∇A−12 and ∣V∣12A−12 on Lp(RN,wdx). Our result extends the one of Bui (2010) [14] to signed potentials and treat the case where p≥2. It also gives a weighted version of our earlier results Assaad (2011) [1], Assaad and Ouhabaz (2012) [2] and of the result (Auscher and Ben Ali 2007) [4] to weighted Lebesgue spaces. We study also the boundedness from Lp(RN,wpdx) to Lq(RN,wqdx) of the fractional power A−α/2 and the Lp(RN,wdx)-boundedness of the H∞-functional calculus of A.