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Directional Poincaré inequality on compact Lie groups

Paulo L. Dattori da Silva, André Pedroso Kowacs

TL;DR

This work extends Steinerberger's directional Poincaré inequality from the torus to compact Lie groups by establishing a sharp spectral criterion: the inequality on the subspace orthogonal to the kernel of a left-invariant vector field $Y=\langle \nabla_G,\alpha\rangle$ holds if and only if the smallest nonzero singular value of the symbol satisfies $\lambda_{\min}^{>0}[\sigma_Y(\xi)] \ge C\langle \xi\rangle^{-(\delta-1)}$ for all nontrivial representations $[\xi]$. It further proves that such an inequality is equivalent to the global solvability of the associated Fourier multiplier, linking functional-analytic solvability to spectral gaps of the symbol; this leads to notable corollaries for the torus and $\mathbb{S}^3$, including a universal $\delta=1$ bound on $\mathbb{S}^3$ for left-invariant fields. The paper also analyzes tube-type vector fields on $\mathbb{T}^1\times G$, showing that directional Poincaré inequalities are equivalent to global solvability in this setting, and provides explicit conditions involving $a_0$ and the spectrum of $X$, with sharp results when $a_0$ is irrational non-Liouville. Overall, the results create a spectral–geometric framework connecting directional Poincaré inequalities, global solvability of Fourier multipliers, and hypoellipticity-type behavior on compact manifolds.

Abstract

We extend the directional Poincaré inequality on the torus, introduced by Steinerberger in [Ark. Mat. 54 (2016), pp. 555--569], to the setting of compact Lie groups. We provide necessary and sufficient conditions for the existence of such an inequality based on estimates on the eigenvalues of the global symbol of the corresponding vector field. We also prove that such refinement of the Poincaré inequality holds for a left-invariant vector field on a compact Lie group $G$ if and only if the vector field is globally solvable, and extend this equivalence to tube-type vector fields on $\mathbb{T}^1\times G$.

Directional Poincaré inequality on compact Lie groups

TL;DR

This work extends Steinerberger's directional Poincaré inequality from the torus to compact Lie groups by establishing a sharp spectral criterion: the inequality on the subspace orthogonal to the kernel of a left-invariant vector field holds if and only if the smallest nonzero singular value of the symbol satisfies for all nontrivial representations . It further proves that such an inequality is equivalent to the global solvability of the associated Fourier multiplier, linking functional-analytic solvability to spectral gaps of the symbol; this leads to notable corollaries for the torus and , including a universal bound on for left-invariant fields. The paper also analyzes tube-type vector fields on , showing that directional Poincaré inequalities are equivalent to global solvability in this setting, and provides explicit conditions involving and the spectrum of , with sharp results when is irrational non-Liouville. Overall, the results create a spectral–geometric framework connecting directional Poincaré inequalities, global solvability of Fourier multipliers, and hypoellipticity-type behavior on compact manifolds.

Abstract

We extend the directional Poincaré inequality on the torus, introduced by Steinerberger in [Ark. Mat. 54 (2016), pp. 555--569], to the setting of compact Lie groups. We provide necessary and sufficient conditions for the existence of such an inequality based on estimates on the eigenvalues of the global symbol of the corresponding vector field. We also prove that such refinement of the Poincaré inequality holds for a left-invariant vector field on a compact Lie group if and only if the vector field is globally solvable, and extend this equivalence to tube-type vector fields on .

Paper Structure

This paper contains 9 sections, 21 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Directional Poincaré inequality on compact Lie groups
  4. Necessary and sufficient conditions
  5. The directional Poincaré inequality on the torus
  6. Directional Poincaré Inequality on the 3-sphere
  7. Global Solvability of Fourier Multipliers on compact Lie groups
  8. Directional Poincaré inequality for tube-type vector fields
  9. Acknowledgements

Key Result

Proposition 2.1

Let $({v}(\xi))_{[\xi]\in\widehat{G}}$ denote a sequence of matrices such that $v(\xi)\in \mathbb{C}^{d_\xi\times d_\xi}$ for every $[\xi]\in\widehat{G}$. Then $v(\cdot)=\widehat{f}(\cdot)$ for a smooth function $f\in C^\infty(G)$ if and only if for every $N>0$, there exists $C_N>0$ such that for every $[\xi]\in\widehat{G}$. Moreover, $v(\cdot)=\widehat{u}(\cdot)$ for a distribution $u\in \mathca

Theorems & Definitions (39)

  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Definition 3.3
  • Remark 1
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • ...and 29 more