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

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 $L_w^p$ if and only if $w$ is in the class $A_{p,\lambda }$. We introduce a new class of Muckenhoupt-type weights ${\tilde{A}}_{p,\lambda }$ 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.