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

This paper aims to study $A_p$ 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 $L^p$ 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 H\"older inequality in the case when the flow measure is locally doubling. We finally show that the logarithm of an $A_p$ weight is in BMO and discuss the connection between $A_p$ weights and quasisymmetric mappings.