Hardy Inequalities for the Heisenberg Laplacian on convex bounded polytopes

TL;DR

This paper establishes a Hardy inequality for the Heisenberg Laplacian on convex polytopes, linking boundary distance with the Carnot-Carathéodory metric, and analyzing the dependence of the constant on boundary hyperplanes.

Contribution

It introduces a Hardy inequality for the Heisenberg Laplacian on convex polytopes, with a novel boundary distance weight and explicit dependence of the constant on boundary hyperplanes.

Findings

Hardy inequality proven for the Heisenberg Laplacian on convex polytopes

Integral weight based on Carnot-Carathéodory boundary distance

Constant depends on the number and orientation of boundary hyperplanes

Abstract

We prove a Hardy-type inequality for the gradient of the Heisenberg Laplacian on open bounded convex polytopes on the first Heisenberg Group. The integral weight of the Hardy inequality is given by the distance function to the boundary measured with respect to the Carnot-Carath\'{e}odory metric. The constant depends on the number of hyperplanes, given by the boundary of the convex polytope, which are not orthogonal to the hyperplane $x_3=0$.