A unified approach to Hardy-type inequalities with Bessel pairs

TL;DR

This paper develops a unified framework for Hardy-type inequalities using Bessel pairs, extending classical results to more general settings and providing explicit maximizers and sharp constants.

Contribution

It introduces a general characterization of Bessel pairs that ensures Hardy inequalities, extending the scope beyond Euclidean spaces and unifying previous results.

Findings

Explicit formulas for maximizing functions

Sharp constants in specific cases

Extension of Hardy inequalities to broader contexts

Abstract

In this paper, we provide suitable characterisations of pairs of weights $(V,W),$ known as Bessel pairs, that ensure the validity of weighted Hardy-type inequalities. The abstract approach adopted here makes it possible to establish such inequalities also going beyond the classical Euclidean setting and also within a more general $L^p$ framework. As a byproduct of our method, we obtain explicit expressions for the maximizing functions and, in certain specific situations, we show that the associated constants are sharp. We emphasise that our approach unifies, generalises and improves several existing results in the literature.