Second order Riesz transforms associated to the Schrِdinger operator for

Let L = − Δ + V be the Schrödinger operator on R n , where V belongs to the class of reverse Hölder weights R H q for some q > max { 2 , n / 2 } . We show that the second order Riesz transforms ∇ 2 L − 1 and V L − 1 are bounded from the Hardy spaces H L p ( R n ) associated to L into L p ( R n ) for 0 < p ⩽ 1 . We show also that the operators ∇ 2 L − 1 map the classical Hardy spaces H p ( R n ) into H p ( R n ) for a restricted range of p.