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Poincaré inequalities on Carnot Groups and spectral gap of Schrödinger operators

Marianna Chatzakou, Serena Federico, Boguslaw Zegarlinski

TL;DR

The paper addresses global Poincaré inequalities on Carnot groups for probability measures defined by densities of a homogeneous quasi-norm, and links these inequalities to spectral gaps of Schrödinger-type operators. It develops the $U$-bounds framework to derive global $q$-Poincaré inequalities from gradient controls on a homogeneous quasi-norm $N$, proving a sufficient condition (Theorem thm.SuffCond2) that yields Poincaré inequalities for measures $\mu_p = Z^{-1} e^{-a N^p}$ when $p \ge 2\gamma$. It further shows there exists a quasi-norm on any Carnot group for which the global Poincaré inequality holds, via a coordinate-change argument (Theorem THM:hypothesis), and derives the spectral gap for the associated operator $\mathcal{L}_p = -\Delta_{\mathbb{G}} + a p N^{p-1} ∇_{\mathbb{G}} N · ∇_{\mathbb{G}}$. The results cover extensive classes of Carnot groups (including step-2 and $H$-type groups) with explicit norms (e.g., Kaplan-type, $N_α$) and provide concrete examples, extending prior work and enabling exponential convergence results for the corresponding Schrödinger-type dynamics.

Abstract

In this work we give a sufficient condition under which the global Poincaré inequality on Carnot groups holds true for a large family of probability measures absolutely continuous with respect to the Lebesgue measure. The density of such probability measure is given in terms of homogeneous quasi-norm on the group. We provide examples to which our condition applies including the most known families of Carnot groups. This, in particular, allows to extend the results in the previous work [CFZ21]. A consequence of our result is that the associated Schrödinger operators have a spectral gap.

Poincaré inequalities on Carnot Groups and spectral gap of Schrödinger operators

TL;DR

The paper addresses global Poincaré inequalities on Carnot groups for probability measures defined by densities of a homogeneous quasi-norm, and links these inequalities to spectral gaps of Schrödinger-type operators. It develops the -bounds framework to derive global -Poincaré inequalities from gradient controls on a homogeneous quasi-norm , proving a sufficient condition (Theorem thm.SuffCond2) that yields Poincaré inequalities for measures when . It further shows there exists a quasi-norm on any Carnot group for which the global Poincaré inequality holds, via a coordinate-change argument (Theorem THM:hypothesis), and derives the spectral gap for the associated operator . The results cover extensive classes of Carnot groups (including step-2 and -type groups) with explicit norms (e.g., Kaplan-type, ) and provide concrete examples, extending prior work and enabling exponential convergence results for the corresponding Schrödinger-type dynamics.

Abstract

In this work we give a sufficient condition under which the global Poincaré inequality on Carnot groups holds true for a large family of probability measures absolutely continuous with respect to the Lebesgue measure. The density of such probability measure is given in terms of homogeneous quasi-norm on the group. We provide examples to which our condition applies including the most known families of Carnot groups. This, in particular, allows to extend the results in the previous work [CFZ21]. A consequence of our result is that the associated Schrödinger operators have a spectral gap.
Paper Structure (9 sections, 13 theorems, 96 equations)

This paper contains 9 sections, 13 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Hörmander sum of squares and Poincaré inequalities
  3. Operators with discrete spectra on Carnot groups and the $U$-bounds method
  4. The class of probability measures for which the spectral gap inequality holds true
  5. Main result and spectral gap
  6. Organisation of the paper
  7. Preliminaries
  8. Sufficient conditions for global Poincaré inequalities: proof of Theorem \ref{['thm.SuffCond2']}
  9. Examples and general results

Key Result

Theorem 1.1

Let $\mathbb G$ be any nilpotent Lie group with the Haar measure $dx$. For any $p \in [1,\infty)$ there exists a constant $P_{0}(r)=P_{0}(r,p)$ such that, for every $x\in \mathbb{G}$ and $f \in C^{\infty}(B_r(x))$, where $f_{B_{r}(x)}:= \frac{1}{|B_{r}(x)|}\int_{B_{r}(x)}f(y)\,dy$ and $B_{r}(x):=\{y \in \mathbb G : d(x,y) \leq r \}$ is the ball of radius $r$ centered at $x$ with respect to the Ca

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Definition 2.1: Homogeneous Lie group on $\mathbb{R}^n$
  • Definition 2.2: $\mathbb{G}$-length of a multi-index
  • Definition 2.3: $\delta_\lambda$-homogeneous function of degree $m$
  • Definition 2.4: $\mathbb{G}$-degree of a polynomial function
  • Proposition 2.5: Smooth $\delta_\lambda$-homogeneous functions
  • Remark 2.6
  • ...and 20 more