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A unified approach to Hardy-type inequalities with Bessel pairs

Lucrezia Cossetti, Lorenzo D'Arca

TL;DR

This work develops a unified, abstract framework for Hardy-type inequalities via Bessel-type weight pairs $(V,W)$ in broad geometric settings, including Euclidean, subelliptic, and Carnot-group contexts. Central to the approach are scalar and vector-valued algebraic identities that, through strategic choices of auxiliary functions, yield Hardy along a fixed direction and a complete Hardy inequality, with equality attained by natural maximizers. The paper then derives a rich tapestry of sharp, high-signal results: d-radial, cylindrical, logarithmic, and Gaussian Hardy inequalities; Hardy bounds in an annulus; and applications to antisymmetric domains and several subelliptic operators (Heisenberg-Greiner, Baouendi-Grushin, Carnot groups). Collectively, these results generalize and unify numerous classical inequalities, providing explicit maximizing functions and sharp constants across diverse geometric-operator settings. The framework offers a versatile toolkit for further extensions in non-Euclidean analysis and PDEs with weighted energies.

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.

A unified approach to Hardy-type inequalities with Bessel pairs

TL;DR

This work develops a unified, abstract framework for Hardy-type inequalities via Bessel-type weight pairs in broad geometric settings, including Euclidean, subelliptic, and Carnot-group contexts. Central to the approach are scalar and vector-valued algebraic identities that, through strategic choices of auxiliary functions, yield Hardy along a fixed direction and a complete Hardy inequality, with equality attained by natural maximizers. The paper then derives a rich tapestry of sharp, high-signal results: d-radial, cylindrical, logarithmic, and Gaussian Hardy inequalities; Hardy bounds in an annulus; and applications to antisymmetric domains and several subelliptic operators (Heisenberg-Greiner, Baouendi-Grushin, Carnot groups). Collectively, these results generalize and unify numerous classical inequalities, providing explicit maximizing functions and sharp constants across diverse geometric-operator settings. The framework offers a versatile toolkit for further extensions in non-Euclidean analysis and PDEs with weighted energies.

Abstract

In this paper, we provide suitable characterisations of pairs of weights 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 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.

Paper Structure

This paper contains 19 sections, 59 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Proof of main results
  3. Application I: $d$-radial Hardy-type inequalities
  4. Classical weighted Hardy inequalities
  5. Hardy inequalities related to $p$-Harmonic distance functions
  6. Sharp Hardy inequalities related to fundamental solutions
  7. Cylindrical Hardy inequalities
  8. Logarithmic Hardy inequalities
  9. Sharp Logarithmic Hardy inequalities
  10. Cylindrical Logarithmic Hardy inequalities
  11. Gaussian Hardy inequalities
  12. Hardy inequalities in an annulus
  13. Application II: Hardy inequality for special domains and antisymmetric functions
  14. Application III: Hardy-type inequalities for some subelliptic operators
  15. Euclidean Setting
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\Omega \subseteq \mathbb{R}^N$ be an open set. Assume $p\geq2$ and consider a vector field $Z \colon \Omega \subseteq \mathbb{R}^N \to \mathbb{R}^h$ and two functions $V(x) \geq 0$ and $W(x)$ such that $W(x) \in L^1_{\text{loc}}(\Omega)$ and $V(x) |Z|^p \in L^1_{\text{loc}}(\Omega)$. Let $\lamb which means Then for all $u \in C^\infty_c(\Omega;\mathbb{C})$ one has Moreover, the function $u=

Theorems & Definitions (105)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Remark 2.1
  • Remark 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm: Hardy lungo Z']}
  • Proposition 2.2
  • Remark 2.3
  • ...and 95 more