Geometric Hardy inequalities on the Heisenberg groups via convexity
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915882105569280 |
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| author | Barbatis, Gerassimos Chatzakou, Marianna Tertikas, Achilles |
| author_facet | Barbatis, Gerassimos Chatzakou, Marianna Tertikas, Achilles |
| contents | We prove $L^p$-Hardy inequalities with distance to the boundary for domains in the Heisenberg group ${\mathbb{H}}^n$, $n\geq 1$. Our results are based on a certain geometric condition. This is first implemented for the Euclidean distance in certain non-convex domains. It is then implemented for the distance defined by the gauge quasi-norm related to the fundamental solution of the horizontal Laplacian when the domain is a half-space or a convex polytope. Finally it is implemented for the Carnot-Carathéodory distance on half-spaces and arbitrary bounded convex domains of ${\mathbb{H}}^n$. In all cases the constant $((p-1)/p)^p$ is obtained. In the more general context of a stratified Lie group of step two we study the superharmonicity and the weak $H$-concavity of the Euclidean distance to the boundary, thus obtaining a proof of the $L^p$-Hardy inequality on convex domains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08383 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric Hardy inequalities on the Heisenberg groups via convexity Barbatis, Gerassimos Chatzakou, Marianna Tertikas, Achilles Analysis of PDEs 35R03, 26D10 (Primary) 35A23, 35H20, 35J75 (Secondary) We prove $L^p$-Hardy inequalities with distance to the boundary for domains in the Heisenberg group ${\mathbb{H}}^n$, $n\geq 1$. Our results are based on a certain geometric condition. This is first implemented for the Euclidean distance in certain non-convex domains. It is then implemented for the distance defined by the gauge quasi-norm related to the fundamental solution of the horizontal Laplacian when the domain is a half-space or a convex polytope. Finally it is implemented for the Carnot-Carathéodory distance on half-spaces and arbitrary bounded convex domains of ${\mathbb{H}}^n$. In all cases the constant $((p-1)/p)^p$ is obtained. In the more general context of a stratified Lie group of step two we study the superharmonicity and the weak $H$-concavity of the Euclidean distance to the boundary, thus obtaining a proof of the $L^p$-Hardy inequality on convex domains. |
| title | Geometric Hardy inequalities on the Heisenberg groups via convexity |
| topic | Analysis of PDEs 35R03, 26D10 (Primary) 35A23, 35H20, 35J75 (Secondary) |
| url | https://arxiv.org/abs/2503.08383 |