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Weighted Poincaré inequality and Hardy improvements related to some degenerate elliptic differential operators

Lorenzo D'Arca

TL;DR

The paper develops a unified approach to sharp weighted Poincaré inequalities and derives $L^p$ Hardy-type improvements with remainder terms in Euclidean and non-Euclidean geometries. By working with a general family of vector fields $X_i$ and a degree-one homogeneous distance $d$ in a space with homogeneous dimension $Q$, it identifies a radial eigenfunction $\varphi$ whose first zero $\nu_1(p,\theta)$ dictates the optimal constants. It then furnishes two Hardy refinements, Type I and Type II, featuring constants $\lambda_p$ and $z_0$ and explicit forms in special cases like $p=2$, and demonstrates how the results specialize to classical settings as well as to sub-elliptic operators such as Baouendi-Grushin, Heisenberg, and Carnot-group Laplacians. Overall, the work provides robust, geometry-aware tools for spectral inequalities and PDE analysis across a broad class of spaces, with potential applications to eigenvalue problems and stability estimates on groups and degenerate operators.

Abstract

In this paper, we characterize the sharp constant and maximizing functions for weighted Poincaré inequalities. These results lead to refinements of Hardy's inequality obtained by adding remainder terms involving \(L^p\) norms. We use techniques that avoid symmetric rearrangement argument, simplifying the analysis of these inequalities in both Euclidean and non-Euclidean contexts. Specifically, this method applies to a variety of settings, such as the Heisenberg group, various Carnot groups and operators expressed as sums of squares of vector fields. Significant examples include the Heisenberg-Greiner operator and the Baouendi-Grushin operator.

Weighted Poincaré inequality and Hardy improvements related to some degenerate elliptic differential operators

TL;DR

The paper develops a unified approach to sharp weighted Poincaré inequalities and derives Hardy-type improvements with remainder terms in Euclidean and non-Euclidean geometries. By working with a general family of vector fields and a degree-one homogeneous distance in a space with homogeneous dimension , it identifies a radial eigenfunction whose first zero dictates the optimal constants. It then furnishes two Hardy refinements, Type I and Type II, featuring constants and and explicit forms in special cases like , and demonstrates how the results specialize to classical settings as well as to sub-elliptic operators such as Baouendi-Grushin, Heisenberg, and Carnot-group Laplacians. Overall, the work provides robust, geometry-aware tools for spectral inequalities and PDE analysis across a broad class of spaces, with potential applications to eigenvalue problems and stability estimates on groups and degenerate operators.

Abstract

In this paper, we characterize the sharp constant and maximizing functions for weighted Poincaré inequalities. These results lead to refinements of Hardy's inequality obtained by adding remainder terms involving norms. We use techniques that avoid symmetric rearrangement argument, simplifying the analysis of these inequalities in both Euclidean and non-Euclidean contexts. Specifically, this method applies to a variety of settings, such as the Heisenberg group, various Carnot groups and operators expressed as sums of squares of vector fields. Significant examples include the Heisenberg-Greiner operator and the Baouendi-Grushin operator.
Paper Structure (15 sections, 37 theorems, 188 equations)

This paper contains 15 sections, 37 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. The main results
  3. Weighted Poincaré inequality
  4. Some Special Cases
  5. The case $\theta = 1$.
  6. The case $p=2$.
  7. Improved Hardy inequality
  8. Poincaré inequalities and Hardy improvements for some sub-elliptic operators
  9. Euclidean Laplacian
  10. Baouendi-Grushin Operator
  11. Heisenberg Sub-Laplacian
  12. The Heisenberg-Greiner operator
  13. Sub-Laplacian on Homogeneous Carnot group
  14. Appendix
  15. Appendix

Key Result

Proposition 1.1

Let $\theta \geq 1$. The problem admits a unique solution $\varphi \in C^1[0,+\infty)$ with $r^{\theta-1}\phi_p(\varphi') \in C^1[0,+\infty)$. Furthermore, this solution is oscillatory, meaning for every $r > 0$ there exists a $t > r$ such that $\varphi(t) = 0$, and each zero is simple (if $\varphi(z_0) = 0$, then $\varphi'(z_0) \n

Theorems & Definitions (69)

  • Proposition 1.1
  • Remark 1.1
  • Proposition 1.2
  • Remark 1.2
  • Theorem 1.1: Weighted Poincaré inequality
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.2: Hardy Improvement of type I
  • Remark 1.5
  • Theorem 1.3: Hardy Improvement of Type II
  • ...and 59 more