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Spectral gap bounds on H-type groups

Marco Carfagnini, Maria Gordina

Abstract

In this note we provide bounds on the spectral gap for the Dirichlet sub-Laplacians on $H$-type groups. We use probabilistic techniques and in particular small deviations of the corresponding hypoelliptic Brownian motion.

Spectral gap bounds on H-type groups

Abstract

In this note we provide bounds on the spectral gap for the Dirichlet sub-Laplacians on -type groups. We use probabilistic techniques and in particular small deviations of the corresponding hypoelliptic Brownian motion.
Paper Structure (7 sections, 7 theorems, 70 equations)

This paper contains 7 sections, 7 theorems, 70 equations.

Table of Contents

  1. Introduction and main result
  2. Geometric background
  3. Carnot groups
  4. $H$-type groups
  5. Proof of the main result
  6. A probabilistic lemma
  7. Spectral bounds

Key Result

Theorem 1

Let $\mathfrak{g}$ be an $H$-type Lie algebra with center $\mathfrak{z} \cong \mathbb{R}^{n}$. Let $\mathbb{G} \cong \mathbb{R}^{m} \times \mathbb{R}^{n}$ be the corresponding $H$-type group with the homogeneous norm $\vert \cdot \vert$ and the sub-Laplacian $\Delta_{\mathbb{G}}$. Then where $\lambda_{1} = \lambda_{1} (m,n)$ is the spectral gap of $-\frac{1}{2} \Delta_{\mathbb{G}}$ restricted to

Theorems & Definitions (22)

  • Theorem : Theorem \ref{['thm.main']}
  • Definition 2.1: Carnot groups
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5: Corollary 1 in Kaplan1980
  • Example 2.6
  • Example 2.7
  • Remark 2.8
  • Theorem 2.9
  • ...and 12 more