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

TL;DR

This paper extends classical $A_p$ weight theory to Schr"odinger operators with potentials in reverse H"older classes, establishing new bounds and links with BMO spaces, leading to improved quantitative estimates for associated operators.

Contribution

It introduces the $A_p^L$ weight class for Schr"odinger operators, extending classical results, and provides sharper bounds for related maximal and fractional integral operators.

Findings

Established the 'exp--log' link between $A_p^L$ and $BMO_L$.

Proved quantitative bounds for maximal functions and heat semigroup.

Provided improved constants for fractional integral operators.

Abstract

Suppose $L=-\Delta+V$ is a Schr\"odinger operator on $\mathbb{R}^n$ with a potential $V$ belonging to certain reverse H\"older class $RH_\sigma$ with $\sigma\geq n/2$. 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 establish the "exp--log" link between $A_p^L$ and $BMO_L$ (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 $L$.