Bellman function for Hardy's inequality over dyadic trees

TL;DR

This paper applies the Bellman function technique to characterize measures for which Hardy's inequality holds on dyadic trees, providing explicit extremal examples and an interpretation via stochastic control.

Contribution

It introduces a Bellman function approach to Hardy's inequality on dyadic trees, establishing optimal constants and explicit extremal measures.

Findings

Characterization of measures satisfying Hardy's inequality on dyadic trees

Explicit extremal family of maps demonstrating optimality

Interpretation of Bellman function through stochastic optimal control

Abstract

In this article we use the Bellman function technique to characterize the measures for which the weighted Hardy's inequality holds on dyadic trees. We enunciate the (dual) Hardy's inequality over the dyadic tree and we use the associated "Burkholder-type" function to find the main inequality required in the proof of the result. We enunciate a "Bellman-type" function satisfying the required properties and we use the Bellman function technique to prove the main result of this article. We prove the optimality of the associated constant with an explicit example of an extremal family of maps. We also give an explicit interpretation of the corresponding Bellman function in terms of the theory of stochastic optimal control.