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arXiv · 2603.04086

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

Lorenzo d'Arca·Luca Fanelli·Valentina Franceschi·Dario Prandi
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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{\`a}nyi gauge and the Carnot--Carath{\'e}odory distance, and we extend the results to non-isotropic step-two structures through a generalized Kor{\`a}nyi-type homogeneous norm.

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Lorenzo d'Arca, Luca Fanelli, Valentina Franceschi, Dario Prandi. 2026-03-04. Unweighted Hardy Inequalities on the Heisenberg Group and in Step-Two Carnot Groups. https://arxiv.org/abs/2603.04086

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