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Geometric Hardy inequalities on the Heisenberg groups via convexity

Gerassimos Barbatis, Marianna Chatzakou, Achilles Tertikas

Abstract

We prove $L^p$-Hardy inequalities with distance to the boundary for domains in the Heisenberg group ${\mathbb{H}}^n$, $n\geq 1$. Our results are based on a certain geometric condition. This is first implemented for the Euclidean distance in certain non-convex domains. It is then implemented for the distance defined by the gauge quasi-norm related to the fundamental solution of the horizontal Laplacian when the domain is a half-space or a convex polytope. Finally it is implemented for the Carnot-Carathéodory distance on half-spaces and arbitrary bounded convex domains of ${\mathbb{H}}^n$. In all cases the constant $((p-1)/p)^p$ is obtained. In the more general context of a stratified Lie group of step two we study the superharmonicity and the weak $H$-concavity of the Euclidean distance to the boundary, thus obtaining a proof of the $L^p$-Hardy inequality on convex domains.

Geometric Hardy inequalities on the Heisenberg groups via convexity

Abstract

We prove -Hardy inequalities with distance to the boundary for domains in the Heisenberg group , . Our results are based on a certain geometric condition. This is first implemented for the Euclidean distance in certain non-convex domains. It is then implemented for the distance defined by the gauge quasi-norm related to the fundamental solution of the horizontal Laplacian when the domain is a half-space or a convex polytope. Finally it is implemented for the Carnot-Carathéodory distance on half-spaces and arbitrary bounded convex domains of . In all cases the constant is obtained. In the more general context of a stratified Lie group of step two we study the superharmonicity and the weak -concavity of the Euclidean distance to the boundary, thus obtaining a proof of the -Hardy inequality on convex domains.

Paper Structure

This paper contains 9 sections, 23 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. two general results on stratified groups
  3. Geometric Hardy inequalities on the Heisenberg group
  4. Hardy inequalities with respect to the Euclidean distance: remarks and counterexamples
  5. Hardy inequalities with respect to the gauge pseudodistance
  6. Hardy inequalities with respect to the Carnot-Carathéodory distance
  7. Hardy inequalities on stratified groups of step two
  8. On the distance function from the boundary of bounded convex domains in stratified groups
  9. Hardy inequalities with respect to the Euclidean distance

Key Result

Theorem 1

Let $G$ be a stratified group and let $\Omega\subset G$ be open and connected. Let $p>1$ and let $d: \Omega \rightarrow (0,\infty)$ be a positive, locally CC-Lipschitz function. (a) Assume that where the inequality is understood in the distributional sense. Then (b) Assume that there exist $x_0 \in \partial \Omega$ and two neighbourhoods $A,A'$ of $x_0$ in $G$ with $A'\subset\subset A$ and such

Theorems & Definitions (37)

  • Theorem 1
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • Remark 1
  • Theorem 5
  • Remark 2
  • Lemma 6
  • Proposition 7
  • ...and 27 more