Weighted mixed inequalities for commutators of Schr\"odinger type operators

TL;DR

This paper establishes weighted mixed inequalities for commutators of Schr"odinger-type operators, extending previous results and involving novel classes of weights related to a critical radius function.

Contribution

It introduces new weighted mixed inequalities for commutators of Schr"odinger-Calderón-Zygmund operators with BMO symbols, generalizing prior estimates and involving $A_p^ ho$ weights.

Findings

Derived estimates for commutators of Schr"odinger-Calderón-Zygmund operators.

Extended mixed inequality results to include $A_p^ ho$ weights.

Generalized previous inequalities for Schr"odinger type operators.

Abstract

We obtain weighted mixed inequalities for the first order commutator of singular integral operators in the Schr\"odinger setting. Concretely, for $0<\delta\leq 1$ we give estimates of commutators of Schr\"odinger-Calder\'on-Zygmund operators of $(s,\delta)$ type with $1<s\leq \infty$, and $\text{BMO}(\rho)$ symbols associated to a critical radious function $\rho$. Our results generalizes some previous estimates about mixed inequalities for Schr\"odinger type operators. We also deal with $A_p^\rho$ weights, which can be understood as a perturbation of the $A_p$ Muckenhoupt classes by means of function $\rho$.