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A unified approach to $L^p$ Hardy and Rellich-type inequalities in Euclidean and non-Euclidean settings

Lorenzo D'Arca

TL;DR

The paper develops a unified, constructive framework for $L^p$ Hardy and Rellich inequalities applicable to a wide class of subelliptic divergence-type operators, both in Euclidean and non-Euclidean settings. Central to the method is a nonlinear algebraic identity with a nonnegative weight that yields explicit maximizing sequences and sharp constants, independent of an underlying Euler operator. Under structural conditions on a distance function $d$ and the operator $\mathcal{L}$, the authors derive weighted Hardy and Rellich inequalities and identify $d$-radial extremals; sharpness is established via smooth cut-offs and explicit asymptotics. The framework applies to the Euclidean Laplacian, Heisenberg--Greiner, Baouendi--Grushin, and Carnot groups (including polarizable groups), providing a unified, degree-two-homogeneous approach that extends beyond homogeneous-group settings and strengthens tools for PDE existence and nonexistence analyses in subelliptic contexts.

Abstract

We present a unified and concise method for establishing L^p Hardy and Rellich inequalities for a broad class of subelliptic operators of divergence type. The approach, based on a fundamental algebraic identity, provides explicit control on maximizing sequences and yields sharp constants in several significant cases. It applies beyond the Euclidean framework, covering the Heisenberg and Carnot group settings, and extends to a variety of subelliptic operators such as the Heisenberg-Greiner and Baouendi-Grushin operators.

A unified approach to $L^p$ Hardy and Rellich-type inequalities in Euclidean and non-Euclidean settings

TL;DR

The paper develops a unified, constructive framework for Hardy and Rellich inequalities applicable to a wide class of subelliptic divergence-type operators, both in Euclidean and non-Euclidean settings. Central to the method is a nonlinear algebraic identity with a nonnegative weight that yields explicit maximizing sequences and sharp constants, independent of an underlying Euler operator. Under structural conditions on a distance function and the operator , the authors derive weighted Hardy and Rellich inequalities and identify -radial extremals; sharpness is established via smooth cut-offs and explicit asymptotics. The framework applies to the Euclidean Laplacian, Heisenberg--Greiner, Baouendi--Grushin, and Carnot groups (including polarizable groups), providing a unified, degree-two-homogeneous approach that extends beyond homogeneous-group settings and strengthens tools for PDE existence and nonexistence analyses in subelliptic contexts.

Abstract

We present a unified and concise method for establishing L^p Hardy and Rellich inequalities for a broad class of subelliptic operators of divergence type. The approach, based on a fundamental algebraic identity, provides explicit control on maximizing sequences and yields sharp constants in several significant cases. It applies beyond the Euclidean framework, covering the Heisenberg and Carnot group settings, and extends to a variety of subelliptic operators such as the Heisenberg-Greiner and Baouendi-Grushin operators.
Paper Structure (13 sections, 18 theorems, 117 equations)

This paper contains 13 sections, 18 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Organization of the Paper
  3. Auxiliary tools
  4. Algebraic Identities
  5. Smooth cut-off functions
  6. Proof of the main results
  7. Validity of the weighted Hardy and Rellich inequalities
  8. Sharpness of the weighted Hardy and Rellich inequalities
  9. Applications and Model Cases
  10. The Euclidean Laplacian
  11. The Heisenberg--Greiner operator
  12. The Baouendi--Grushin operator
  13. Homogeneous Carnot groups

Key Result

Theorem 1.1

Let $p \geq 2$ and $\beta \in \mathbb{R}$ be such that Then, for every $\theta \in \mathbb{R}$ and $u \in C_c^{\infty}(\mathbb{R}^{N} \!\setminus\! \{0\})$, the following inequality holds: Moreover, if there exists a positive constant $\gamma_p$ such that then the constant on the left-hand side is sharp, and the (non-admissible) extremal function is, up to a multiplicative constant,

Theorems & Definitions (34)

  • Example 1: Euclidean Laplacian
  • Theorem 1.1: Weighted Hardy inequality
  • Theorem 1.2: Weighted Rellich inequality
  • Remark 1.1
  • Proposition 2.1: Fundamental $L^p$ identity
  • proof
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Corollary 2.1
  • ...and 24 more