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Extremizer Stability of Higher-order Hardy-Rellich inequalities for Baouendi--Grushin vector fields

Avas Banerjee, Riju Basak, Prasun Roychowdhury

TL;DR

This work develops a comprehensive treatment of Hardy- and Rellich-type inequalities in radial Baouendi–Grushin spaces, revealing an identity that equates subcritical and critical Hardy forms and establishing extremizer stability. It constructs higher-order radial Hardy–Rellich inequalities with sharp constants, and provides exact $L^2$ remainder identities, all within a polar-coordinate Grushin framework augmented by spherical harmonics. By leveraging two-weight Hardy inequalities and weighted Caffarelli–Kohn–Nirenberg inequalities, the authors refine both subcritical and critical forms and illuminate the role of radial symmetry versus non-radial behavior. The results advance understanding of degenerate subelliptic operators and offer tools for PDE analysis on Grushin-type geometries, with precise stability and remainder structures that could inform future spectral and geometric applications.

Abstract

In this paper, we improve the $L^p$-Rellich and Hardy-Rellich inequalities in the setting of radial Baouendi-Grushin vector fields. We establish an identity relating the subcritical and critical Hardy inequalities, thereby demonstrating their equivalence. Moreover, we obtain improved versions of these inequalities via an analysis of extremizer stability. In the higher-order setting, we derive Hardy-Rellich type inequalities involving all radial operators in the Grushin framework and prove that all resulting constants are sharp. Finally, for the $L^2$-higher-order cases, we compute exact remainder terms by establishing identities rather than inequalities.

Extremizer Stability of Higher-order Hardy-Rellich inequalities for Baouendi--Grushin vector fields

TL;DR

This work develops a comprehensive treatment of Hardy- and Rellich-type inequalities in radial Baouendi–Grushin spaces, revealing an identity that equates subcritical and critical Hardy forms and establishing extremizer stability. It constructs higher-order radial Hardy–Rellich inequalities with sharp constants, and provides exact remainder identities, all within a polar-coordinate Grushin framework augmented by spherical harmonics. By leveraging two-weight Hardy inequalities and weighted Caffarelli–Kohn–Nirenberg inequalities, the authors refine both subcritical and critical forms and illuminate the role of radial symmetry versus non-radial behavior. The results advance understanding of degenerate subelliptic operators and offer tools for PDE analysis on Grushin-type geometries, with precise stability and remainder structures that could inform future spectral and geometric applications.

Abstract

In this paper, we improve the -Rellich and Hardy-Rellich inequalities in the setting of radial Baouendi-Grushin vector fields. We establish an identity relating the subcritical and critical Hardy inequalities, thereby demonstrating their equivalence. Moreover, we obtain improved versions of these inequalities via an analysis of extremizer stability. In the higher-order setting, we derive Hardy-Rellich type inequalities involving all radial operators in the Grushin framework and prove that all resulting constants are sharp. Finally, for the -higher-order cases, we compute exact remainder terms by establishing identities rather than inequalities.
Paper Structure (7 sections, 26 theorems, 201 equations)

This paper contains 7 sections, 26 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Preliminary concepts
  3. Hardy-type inequalities
  4. Hardy--Rellich and Rellich inequalities
  5. Interplay of Critical and Subcritical Hardy inequalities
  6. Extremizer Stability Analysis
  7. Higher order Hardy--Rellich identity

Key Result

Theorem 1.1

Let $G = \mathbb{R}^{n+1}$ and $G' = \mathbb{R}^{m+1}$ be the Grushin spaces. Consider points $(x,t) \in \mathbb{R}^{n+1}$ and $(y,s) \in \mathbb{R}^{m+1}$ in these spaces, and let $\varrho$ and $\tilde{\varrho}$ denote the respective homogeneous norms. Assume that $Q = n + 2 > Q' = m + 2 \geq 3$, a where $\omega_n$ and $\omega_m$ denote the surface measures of the unit spheres in $\mathbb{R}^n$ a

Theorems & Definitions (57)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Theorem 1.3
  • Remark 1.2
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2: FS08
  • Lemma 2.3
  • ...and 47 more