$A_p$ weights and Quantitative Estimates in the Schr\"odinger Setting

Suppose $L=-\mathrm{\Delta }+V$ is a Schr"odinger operator on ${\mathbb{R}}^n$ with a potential $V$ belonging to certain reverse H"older class $R{H}_{\sigma }$ with $\sigma \ge n\mathrm{/}2$. 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 establish the “exp–log” link between $A_p^L$ and $BM{O}_L$ (the BMO space associated with $L$), which is the first extension of the classical result to a setting beyond the Laplace operator. Second, we 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 $A_{p,q}\subset A_{p,q}^L$, 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 $L$.