Muckenhoupt-Type Weights and Quantitative Weighted Estimate in the Bessel Setting

TL;DR

This paper introduces a new class of weights in the Bessel setting to better understand the boundedness of singular integrals, and provides quantitative estimates for these bounds.

Contribution

It defines a new Muckenhoupt-type weight class in the Bessel setting and establishes weighted boundedness, compactness, and endpoint estimates for related operators.

Findings

New weight class $ ilde{A}_{p, u}$ characterizes boundedness of Hardy--Littlewood maximal operators.

Weighted $L^p$ boundedness and compactness of Riesz commutators are proved.

Quantitative bounds for weighted estimates are established.

Abstract

Part of the intrinsic structure of singular integrals in the Bessel setting is captured by Muckenhoupt-type weights. Anderson--Kerman showed that the Bessel Riesz transform is bounded on weighted $L^p_w$ if and only if $w$ is in the class $A_{p,\lambda}$. We introduce a new class of Muckenhoupt-type weights $\widetilde A_{p,\lambda}$ in the Bessel setting, which is different from $A_{p,\lambda}$ but characterizes the weighted boundedness for the Hardy--Littlewood maximal operators. We also establish the weighted $L^p$ boundedness and compactness, as well as the endpoint weak type boundedness of Riesz commutators. The quantitative weighted bound is also established.