A note on fractional type integrals in the Schr\"{o}dinger setting

TL;DR

This paper establishes new weighted inequalities for fractional integrals related to Schr"{o}dinger operators, including endpoint bounds and mixed weak type inequalities, extending the understanding of these operators in weighted function spaces.

Contribution

The paper introduces the first quantitative weighted endpoint bounds for fractional integrals in the Schr"{o}dinger setting and generalizes to mixed weak type inequalities.

Findings

Quantitative $A_{p,q}^{ ext{L}}$ estimates for fractional integrals

First weighted endpoint bounds for Schr"{o}dinger fractional integrals

Extension to weighted mixed weak type inequalities

Abstract

Assume $\mathcal{L}=-\Delta+V$ is a Schr\"{o}dinger operator on $\mathbb{R}^d$, where $V$ belongs to certain reverse H\"{o}lder class $RH_\sigma$ with $\sigma\geq d/2$. We consider the class of $A_{p,q}$ weights associated to $\mathcal{L}$, denoted by $A_{p,q}^{\mathcal{L}}(\mathbb{R}^d)$, which include the classical Muckenhoupt $A_{p,q}(\mathbb{R}^d)$ weights. We obtain the quantitative $A_{p,q}^{\mathcal{L}}(\mathbb{R}^d)$ estimates for fractional integrals associated to the Schr\"{o}dinger operator. Particularly, the quantitative weighted endpoint bound for fractional integrals associated to the Schr\"{o}dinger operator is first established, which was missing in the literature of Li et al. \cite{LRW}. Moreover, we generalize weighted endpoint inequalities to weighted mixed weak type inequalities for fractional type integrals in the Schr\"{o}dinger setting.