Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Unweighted Hardy Inequalities on the Heisenberg Group and in Step-Two Carnot Groups

Lorenzo d'Arca, Luca Fanelli, Valentina Franceschi, Dario Prandi

Abstract

We establish unweighted Hardy-type inequalities on step-two Carnot groups with one-dimensional vertical layer, with explicit lower bounds for the optimal Hardy constant. The approach is based on a quantitative integration-by-parts mechanism that replaces the non-horizontal Euler vector field by a suitably constructed horizontal vector field with controlled norm. As applications, we obtain fully explicit bounds in the Heisenberg group for both the Kor{à}nyi gauge and the Carnot--Carath{é}odory distance, and we extend the results to non-isotropic step-two structures through a generalized Kor{à}nyi-type homogeneous norm.

Unweighted Hardy Inequalities on the Heisenberg Group and in Step-Two Carnot Groups

Abstract

We establish unweighted Hardy-type inequalities on step-two Carnot groups with one-dimensional vertical layer, with explicit lower bounds for the optimal Hardy constant. The approach is based on a quantitative integration-by-parts mechanism that replaces the non-horizontal Euler vector field by a suitably constructed horizontal vector field with controlled norm. As applications, we obtain fully explicit bounds in the Heisenberg group for both the Kor{à}nyi gauge and the Carnot--Carath{é}odory distance, and we extend the results to non-isotropic step-two structures through a generalized Kor{à}nyi-type homogeneous norm.
Paper Structure (10 sections, 15 theorems, 175 equations)

This paper contains 10 sections, 15 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Techniques
  3. Main results
  4. Structure of the paper
  5. Preliminaries
  6. Proof of Theorem \ref{['thm:-Hardy-lungo-Z_d']}
  7. Unweighted Hardy inequality
  8. The Heisenberg group
  9. The non-isotropic case
  10. Product of Heisenberg Groups

Key Result

Theorem 1.1

Let $p \geq 2$, $\theta \in \mathbb{R}$, and let $d$ be a regular positive homogeneous function on $\mathcal{G}$. Then, for every $u \in C_c^\infty(\mathcal{G} \setminus \{0\})$, the following Hardy-type inequality holds Here, the vector field $Z_d$ is defined by If, additionally, $d$ is such that $\langle z,B^{-1}\nabla_z d\rangle=0$ for all $z\in \mathcal{G}\setminus\{0\}$, then inequality eq:

Theorems & Definitions (30)

  • Definition 1
  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • ...and 20 more