Riesz transforms, fractional power and functional calculus of Schrِdinger operators on weighted -spaces

We consider the Schrödinger operator A = − Δ + V + − V − on L p ( R N , w d x ) where N ≥ 3 , and w is a weight in some Muckenhoupt class. We study the boundedness of Riesz transform type operators ∇ A − 1 2 and ∣ V ∣ 1 2 A − 1 2 on L p ( R N , w d x ) . 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 L p ( R N , w p d x ) to L q ( R N , w q d x ) of the fractional power A − α / 2 and the L p ( R N , w d x ) -boundedness of the H ∞ -functional calculus of A .