Weighted estimates for Hardy-Littlewood maximal functions on Harmonic $NA$ groups

Our aim in this article is to study the weighted boundedness of the centered Hardy-Littlewood maximal operator in Harmonic $NA$ groups. Following Ombrosi et al. \cite{ORR}, we define a suitable notion of $A_p$ weights, and for such weights, we prove the weighted $L^p$-boundedness of the maximal operator. Furthermore, as an endpoint case, we prove a variant of the Fefferman-Stein inequality, from which vector-valued maximal inequality has been established. We also provide various examples of weights to substantiate many aspects of our results. In particular, we have shown certain spherical functions of the Harmonic $NA$ group constitute examples of $A_p$ weights. The purely exponential volume growth property of the Harmonic $NA$ group has played a crucial role in our proofs.