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

Abstract

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