Unweighted Hardy Inequalities on the Heisenberg Group and in Step-Two Carnot Groups
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912942620934144 |
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| author | d'Arca, Lorenzo Fanelli, Luca Franceschi, Valentina Prandi, Dario |
| author_facet | d'Arca, Lorenzo Fanelli, Luca Franceschi, Valentina Prandi, Dario |
| contents | 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{à}nyi gauge and the Carnot--Carath{é}odory distance, and we extend the results to non-isotropic step-two structures through a generalized Kor{à}nyi-type homogeneous norm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_04086 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unweighted Hardy Inequalities on the Heisenberg Group and in Step-Two Carnot Groups d'Arca, Lorenzo Fanelli, Luca Franceschi, Valentina Prandi, Dario Analysis of PDEs Functional Analysis 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{à}nyi gauge and the Carnot--Carath{é}odory distance, and we extend the results to non-isotropic step-two structures through a generalized Kor{à}nyi-type homogeneous norm. |
| title | Unweighted Hardy Inequalities on the Heisenberg Group and in Step-Two Carnot Groups |
| topic | Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2603.04086 |