### Abstract

Suppose

*L*= -Δ+V is a Schrödinger operator on ℝ^{n}with a potential V belonging to certain reverse Hölder class*RH*with σ ≥ n/ 2. The aim of this paper is to study the_{σ}*A*weights associated to_{p}*L*, denoted by*A*, which is a larger class than the classical Muckenhoupt_{p}^{L}*A*weights. We first prove the quantitative_{p}*A*bound for the maximal function and the maximal heat semigroup associated to_{p}^{L}*L*. Then we further provide the quantitative*A*bound for the fractional integral operator associated to_{p,q}^{L}*L*. We point out that all these quantitative bounds are known before in terms of the classical*A*constant. However, since_{p,q}*A*⊂_{p,q }*A*, the_{p,q}^{L}*A*constants are smaller than_{p,q}^{L}*A*constant. Hence, our results here provide a better quantitative constant for maximal functions and fractional integral operators associated to_{p}_{,}_{q}*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*and_{p}^{L}*B M O*(the BMO space associated with_{L}*L*), and show that for w ∈*A*, log w is in_{p}^{L}*B M O*, and that the reverse is not true in general._{L}Original language | English |
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Pages (from-to) | 259-283 |

Number of pages | 25 |

Journal | Mathematische Zeitschrift |

Volume | 293 |

Issue number | 1-2 |

Early online date | 14 Nov 2018 |

DOIs | |

Publication status | Published - Oct 2019 |

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### Keywords

- Schrödinger operator
- Weighted inequalities
- Fractional integral operator

### Cite this

Li, J., Rahm, R., & Wick, B. D. (2019).

*A*weights and quantitative estimates in the Schrödinger setting._{p}*Mathematische Zeitschrift*,*293*(1-2), 259-283. https://doi.org/10.1007/s00209-018-2172-4