In this work we obtain boundedness on weighted Lebesgue spaces on R d of the semi-group maximal function, Riesz transforms, fractional integrals and g-function associated to the Schrödinger operator − + V , where V satisfies a reverse Hölder inequality with exponent greater than d/2. We consider new classes of weights that locally behave as Muckenhoupt's weights and actually include them. The notion of locality is defined by means of the critical radius function of the potential V given in Shen (1995) [8].

In this work we obtain boundedness on L p , for 1 < p < ∞, of commuta-where T is any of the Riesz transforms or their conjugates associated to the Schrödinger operator − + V with V satisfying an appropriate reverse Hölder inequality. The class where b belongs is larger than the usual BMO. We also obtain a substitute result for p = ∞, under a slightly stronger condition on b.

In this article we obtain boundedness of the operator (− + V ) −α/2 from L p,∞ (w) into weighted bounded mean oscillation type spaces BMO β L (w) under appropriate conditions on the weight w. We also show that these weighted spaces also have a point-wise description for 0 < β < 1. Finally, we study the behaviour of the operator (− + V ) −α/2 when acting on BMO β L (w).