Muckenhoupt-type weights and quantitative weighted estimates 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 \(L^p_w\) $L_w^p$ if and only if w is in the class \(A_{p,\lambda }\) $A_{p,\lambda }$ . We introduce a new class of Muckenhoupt-type weights \({\widetilde{A}}_{p,\lambda }\) ${\tilde{A}}_{p,\lambda }$ in the Bessel setting, which is different from \(A_{p,\lambda }\) $A_{p,\lambda }$ but characterizes the weighted boundedness for the Hardy–Littlewood maximal operators. We first investigate the quantitative weighted estimates with respect to the new weights \({\widetilde{A}}_{p,\lambda }\) ${\tilde{A}}_{p,\lambda }$ for the sparse operators, the standard one and the one associated to the Bessel BMO space. Then via these sparse operators and the median value technique, we establish the (quantitative) weighted \(L^p\) $L^p$ boundedness and compactness, as well as the endpoint weak type boundedness of Riesz transform commutators.