$$A_p$$ weights on nonhomogeneous trees equipped with measures of exponential growth

This paper aims to study Ap weights in the context of a class of metric measure spaces with exponential volume growth, namely infinite trees with root at infinity equipped with the geodesic distance and flow measures. Our main result is a Muckenhoupt Theorem, which is a characterization of the weights for which a suitable Hardy–Littlewood maximal operator is bounded on the corresponding weighted Lp spaces. We emphasise that this result does not require any geometric assumption on the tree or any condition on the flow measure. We also prove a reverse Hölder inequality in the case when the flow measure is locally doubling. We finally show that the logarithm of an Ap weight is in BMO and discuss the connection between Ap weights and quasisymmetric mappings.