# Hardy-type inequalities for the Carnot-Carath\'eodory distance in the Heisenberg group

## TL;DR

This paper investigates Hardy inequalities in the Heisenberg group, revealing that the optimal constants are smaller than Euclidean cases and that radial inequalities do not hold, highlighting unique sub-Riemannian geometric properties.

## Contribution

The study establishes that the optimal Hardy inequality constant in the Heisenberg group is strictly less than the Euclidean value and shows the non-existence of radial Hardy inequalities, introducing new insights into sub-Riemannian analysis.

## Key findings

- Optimal Hardy constant is less than n^2 in the Heisenberg group.
- Heisenberg group does not support radial Hardy inequalities.
- Constants for non-radial Hardy inequalities are bounded on certain cones.

## Abstract

In this paper we study various Hardy inequalities in the Heisenberg group $\mathbb H^n$, w.r.t. the Carnot-Carath\'eodory distance $\delta$ from the origin. We firstly show that the optimal constant for the Hardy inequality is strictly smaller than $n^2 = (Q-2)^2/4$, where $Q$ is the homogenous dimension. Then, we prove that, independently of $n$, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along $\nabla_{\mathbb H}\delta$. This is in stark contrast with the Euclidean case, where the radial Hardy inequality is equivalent to the standard one, and has the same constant. Motivated by these results, we consider Hardy inequalities for non-radial directions, i.e., directions tangent to the Carnot-Carath\'eodory balls. In particular, we show that the associated constant is bounded on homogeneous cones $C_\Sigma$ with base $\Sigma\subset \mathbb S^{2n}$, even when $\Sigma$ degenerates to a point. This is a genuinely sub-Riemannian behavior, as such constant is well-known to explode for homogeneous cones in the Euclidean space.