Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group

TL;DR

This paper establishes sharp geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group, extending previous results and providing explicit bounds involving Euclidean distances.

Contribution

It generalizes Hardy inequalities to convex domains in the Heisenberg group, including sharp cases for half-spaces, with explicit geometric bounds.

Findings

Proved sharp Hardy inequalities for convex domains in the Heisenberg group.

Extended previous inequalities to more general convex domains.

Provided explicit formulas involving Euclidean distance to the boundary.

Abstract

We prove geometric $L^p$ versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains $\Omega$ in the Heisenberg group $\mathbb{H}^n$, where convex is meant in the Euclidean sense. When $p=2$ and $\Omega$ is the half-space given by $\langle \xi, \nu\rangle > d$ this generalizes an inequality previously obtained by Luan and Yang. For such $p$ and $\Omega$ the inequality is sharp and takes the form \begin{equation} \int_\Omega |\nabla_{\mathbb{H}^n}u|^2 \, d\xi \geq \frac{1}{4}\int_{\Omega} \sum_{i=1}^n\frac{\langle X_i(\xi), \nu\rangle^2+\langle Y_i(\xi), \nu\rangle^2}{\textrm{dist}(\xi, \partial \Omega)^2}|u|^2\, d\xi, \end{equation} where $\textrm{dist}(\, \cdot\,, \partial \Omega)$ denotes the Euclidean distance from $\partial \Omega$.