Two-weight dyadic Hardy's inequalities

TL;DR

This paper surveys and introduces new characterizations of two-weight Hardy's inequalities on infinite trees, including novel proofs and a conformally invariant version, advancing understanding of trace measures and operator compactness.

Contribution

It provides new characterizations, proofs, and a conformally invariant version of two-weight Hardy's inequalities on infinite trees.

Findings

New reverse Hölder inequality for trace measures

Probabilistic proof of Muckenhoupt-Wheeden-Wolff inequality

Characterization of Hardy operator compactness

Abstract

We present various results concerning the two-weight Hardy's inequality on infinite trees. Our main scope is to survey known characterizations (and proofs) for trace measures, as well as to provide some new ones. Also for some of the known characterizations we provide here new proofs. In particular, we obtain a new characterization based on a new reverse H\"older inequality for trace measures, and one based on the well known Muckenhoupt-Wheeden-Wolff inequality, of which we here give a new probabilistic proof. We provide a new direct proof for the so called isocapacitary characterization and a new simple proof, based on a monotonicity argument, for the so called mass-energy characterization. Furthermore, we introduce a conformally invariant version of the two-weight Hardy's inequality, we characterize the compactness of the Hardy operator, we provide a list of open problems and suggest some possible lines of future research.