Two-weight dyadic Hardy's inequalities

We give a survey of results and proofs on two-weight Hardy’s inequalities on infinite trees. Most of the results are already known, but some appear here for the first time. Among the new results that we prove there is the characterization of the compactness of the Hardy operator, a reverse Hölder inequality for trace measures and a simple proof of the characterization of trace measures based on a monotonicity argument. Furthermore we give a probabilistic proof of an inequality due to Wolff. We also provide a list of open problems and suggest some possible lines of future research. 2010 Mathematics Subject Classification. 05C05, 05C63, 31E05, 42B25, 42B35, 31A15, 30C85.

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