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Hardy and Rellich identities and inequalities for Baouendi-Grushin operators via spherical vector fields

Debdip Ganguly, K. Jotsaroop, Prasun Roychowdhury

TL;DR

The paper addresses sharp Hardy, Hardy-Rellich, and Rellich inequalities for the degenerate Baouendi-Grushin operator by developing a unified framework based on $Q$-dimensional Bessel pairs and Garofalo-Shen spherical harmonics. It introduces a spherical vector-field decomposition so that the Grushin operator splits into a radial part and spherical components, enabling abstract Hardy identities and Radial/non-radial Rellich identities with explicit remainder terms. The authors prove an abstract Hardy identity, derive improved Hardy constants in subspaces, and obtain Rellich-type identities with exact deficit terms expressed through spherical eigenvalues, while also establishing symmetrization principles and applications to second-order uncertainty principles and Caffarelli-Kohn-Nirenberg inequalities. These results yield sharper, structurally transparent estimates for Grushin-type operators and open avenues for refined spectral and PDE analyses in hypoelliptic settings.

Abstract

For Baouendi-Grushin vector fields, we prove Hardy, Hardy-Rellich, and Rellich identities and inequalities with sharp constants. Our explicit remainder terms significantly improve than those found in the literature. Our arguments are built on abstract Hardy-Rellich identities involving the Bessel pair along with the use of spherical harmonics developed by Garofalo-Shen [Ann. Inst. Fourier (1994)]. Furthermore, in the spirit of Bez-Machihara-Ozawa [Math. Z (2023)], we construct spherical vector fields corresponding to the Baouendi-Grushin vector fields and prove identities that, in turn, establish optimal Rellich identities, by comparing the Baouendi-Grushin operator with its radial and spherical components. We give alternate proofs of Hardy identities and inequalities with enhanced Hardy constants in some subspaces of the Sobolev space, among other things. Additionally, we compute the deficit involving the $L^2$-norm of the Baouendi-Grushin operator and its radial component with an explicit remainder term, which leads to a comparison of the Baouendi-Grushin operator with its radial components. As a consequence of the main results, new second-order Heisenberg-Pauli-Weyl uncertainty principles and Hydrogen uncertainty principles are also derived. Furthermore, we also derive certain symmetrization principles green corresponding to the Baouendi-Grushin vector fields.

Hardy and Rellich identities and inequalities for Baouendi-Grushin operators via spherical vector fields

TL;DR

The paper addresses sharp Hardy, Hardy-Rellich, and Rellich inequalities for the degenerate Baouendi-Grushin operator by developing a unified framework based on -dimensional Bessel pairs and Garofalo-Shen spherical harmonics. It introduces a spherical vector-field decomposition so that the Grushin operator splits into a radial part and spherical components, enabling abstract Hardy identities and Radial/non-radial Rellich identities with explicit remainder terms. The authors prove an abstract Hardy identity, derive improved Hardy constants in subspaces, and obtain Rellich-type identities with exact deficit terms expressed through spherical eigenvalues, while also establishing symmetrization principles and applications to second-order uncertainty principles and Caffarelli-Kohn-Nirenberg inequalities. These results yield sharper, structurally transparent estimates for Grushin-type operators and open avenues for refined spectral and PDE analyses in hypoelliptic settings.

Abstract

For Baouendi-Grushin vector fields, we prove Hardy, Hardy-Rellich, and Rellich identities and inequalities with sharp constants. Our explicit remainder terms significantly improve than those found in the literature. Our arguments are built on abstract Hardy-Rellich identities involving the Bessel pair along with the use of spherical harmonics developed by Garofalo-Shen [Ann. Inst. Fourier (1994)]. Furthermore, in the spirit of Bez-Machihara-Ozawa [Math. Z (2023)], we construct spherical vector fields corresponding to the Baouendi-Grushin vector fields and prove identities that, in turn, establish optimal Rellich identities, by comparing the Baouendi-Grushin operator with its radial and spherical components. We give alternate proofs of Hardy identities and inequalities with enhanced Hardy constants in some subspaces of the Sobolev space, among other things. Additionally, we compute the deficit involving the -norm of the Baouendi-Grushin operator and its radial component with an explicit remainder term, which leads to a comparison of the Baouendi-Grushin operator with its radial components. As a consequence of the main results, new second-order Heisenberg-Pauli-Weyl uncertainty principles and Hydrogen uncertainty principles are also derived. Furthermore, we also derive certain symmetrization principles green corresponding to the Baouendi-Grushin vector fields.
Paper Structure (9 sections, 24 theorems, 188 equations)

This paper contains 9 sections, 24 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. State of the art and main novelties
  3. Main Results
  4. Preliminaries and functional analysis set-up
  5. Proof of Theorem \ref{['hardy-rell-iden']} and corollaries
  6. Abstract Hardy-Rellich inequality: Proof of Theorem \ref{['r-rell']} and Theorem \ref{['nr-rell']}
  7. Rellich identity and spherical harmonics via spherical derivatives: proof of Theorem \ref{['rellichspherical']}
  8. Symmetrization principle on Grushin space
  9. Applications: Second-order uncertainty Principles and Caffarelli-Kohn-Nirenberg inequalities

Key Result

Theorem 1.1

Let $G$ be the Grushin space with dimension $Q$ with $Q\geq 4$ and $B_R(o)$ denotes the $\varrho-$gauge ball with radius $R$ for $0<R\leq \infty$. If $(V, W)$ is a $Q$-dimensional Bessel pair on $(0,R)$ with positive solution $f$, then for $u\in C_c^\infty(B_R(o)\setminus\{o\})$ there holds and

Theorems & Definitions (59)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.1
  • Corollary 1.2
  • Definition 1.1
  • ...and 49 more