A_p weights and quantitative estimates in the Schrödinger setting

Suppose L= -Δ+V is a Schrödinger operator on ℝⁿ with a potential V belonging to certain reverse Hölder class RH_σ with σ ≥ n/ 2. The aim of this paper is to study the A_p weights associated to L, denoted by A_p^L, which is a larger class than the classical Muckenhoupt A_p weights. We first prove the quantitative A_p^L bound for the maximal function and the maximal heat semigroup associated to L. Then we further provide the quantitative A_p,q^L bound for the fractional integral operator associated to L. We point out that all these quantitative bounds are known before in terms of the classical A_p,q constant. However, since A_p,q ⊂ A_p,q^L, the A_p,q^L constants are smaller than A_p,q constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to L. Next, we prove two–weight inequalities for the fractional integral operator; these have been unknown up to this point. Finally we also have a study on the "exp–log" link between A_p^L and B M O_L (the BMO space associated with L), and show that for w ∈ A_p^L, log w is in B M O_L, and that the reverse is not true in general.