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Optimal subelliptic super-Poincaré and isoperimetric inequalities on stratified Lie groups

Yaozhong W. Qiu

TL;DR

The paper develops an $L^1$-oriented framework to obtain $q$-super-Poincaré and isoperimetric inequalities for exponential-power measures $d\mu = Z^{-1}e^{-N^p}d\xi$ defined via a homogeneous norm $N$ in subelliptic geometries. It relies on $U$-bounds, Hardy inequalities, and $L^1$-Caffarelli–Kohn–Nirenberg tools to derive a $q$-super-Poincaré inequality with growth $\beta_q(\varepsilon) \lesssim \exp(C\varepsilon^{-p(\alpha+1)/(q(p-\alpha-1))})$ for $q\in[1,2]$, and proves isoperimetric bounds $I_\mu \gtrsim \mathcal{U}_r$ with $r = \frac{(\alpha+1)p}{(\alpha+1) + \alpha p}$, which are optimal in several cases. The results are validated across concrete subelliptic settings, including step-two stratified Lie groups, Grushin, and Heisenberg–Greiner operators, as well as an anisotropic Heisenberg group, demonstrating that the tail behavior can be supergaussian while still yielding sharp subgaussian-type isoperimetry. The approach avoids curvature-based assumptions and provides explicit growth rates for the inequalities, with implications for spectral theory and diffusion on non-Euclidean spaces.

Abstract

We prove $q$-super-Poincaré inequalities, $q \in [1, 2]$, for a class of exponential power type probability measures defined in terms of a norm in a number of subelliptic settings, primarily on stratified Lie groups but also in the Grushin and Heisenberg-Greiner settings. Our results include generically optimal isoperimetric inequalities for such probability measures.

Optimal subelliptic super-Poincaré and isoperimetric inequalities on stratified Lie groups

TL;DR

The paper develops an -oriented framework to obtain -super-Poincaré and isoperimetric inequalities for exponential-power measures defined via a homogeneous norm in subelliptic geometries. It relies on -bounds, Hardy inequalities, and -Caffarelli–Kohn–Nirenberg tools to derive a -super-Poincaré inequality with growth for , and proves isoperimetric bounds with , which are optimal in several cases. The results are validated across concrete subelliptic settings, including step-two stratified Lie groups, Grushin, and Heisenberg–Greiner operators, as well as an anisotropic Heisenberg group, demonstrating that the tail behavior can be supergaussian while still yielding sharp subgaussian-type isoperimetry. The approach avoids curvature-based assumptions and provides explicit growth rates for the inequalities, with implications for spectral theory and diffusion on non-Euclidean spaces.

Abstract

We prove -super-Poincaré inequalities, , for a class of exponential power type probability measures defined in terms of a norm in a number of subelliptic settings, primarily on stratified Lie groups but also in the Grushin and Heisenberg-Greiner settings. Our results include generically optimal isoperimetric inequalities for such probability measures.

Paper Structure

This paper contains 11 sections, 13 theorems, 84 equations.

Table of Contents

  1. Introduction and main results
  2. Proof of main results
  3. The case $p > \alpha + 1$
  4. The case $p = \alpha + 1$
  5. Examples
  6. Stratified Lie groups of step two
  7. Grushin and Heisenberg-Greiner operators
  8. The one-dimensional case $n_1 = 1$
  9. An almost $L^1$-Hardy inequality
  10. An $L^1$-Caffarelli-Kohn-Nirenberg inequality
  11. The anisotropic Heisenberg group

Key Result

Theorem 1

Assume $N$ satisfies in the distributional sense for some $\alpha > 0$ and where $\left\lvert x \right\rvert$ is the euclidean norm of a generic point $\xi \in \mathbb{R}^n$ written in the form $\xi = (x, x') \in \mathbb{R}^{n_1} \times \mathbb{R}^{n_2}$ with $n_1 \geq 2$. Assume moreover $\left\lvert x \right\rvert$ satisfies again in the distrib and If there exists a local $1$-super-Poincaré i

Theorems & Definitions (25)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • Remark 1
  • proof
  • Proposition 3
  • Remark 2
  • proof
  • ...and 15 more