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_w^p if and only if w is in the class A_p,λ. We introduce a new class of Muckenhoupt-type weights Ã_p,λ in the Bessel setting, which is different from A_p,λ but characterizes the weighted boundedness for the Hardy–Littlewood maximal operators. We first investigate the quantitative weighted estimates with respect to the new weights Ã_p,λ 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 boundedness and compactness, as well as the endpoint weak type boundedness of Riesz transform commutators.