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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2512.18221 |
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| _version_ | 1866914218370924544 |
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| author | Ma, Shiguang Qing, Jie |
| author_facet | Ma, Shiguang Qing, Jie |
| contents | In this paper we extend the research on potential theory and its geometric applications from Euclidean spaces to homogeneous Carnot groups. We introduce a new approach to use the geometric completeness to estimate the Hausdorff dimension of polar sets of potentials of nonnegative Radon measures for sub-Laplacians in homogeneous Carnot groups. Our approach relies on inequalities that are analogous to the classic integral inequalities about Riesz potentials in Euclidean spaces. Our approach also uses extensions of some of geometric measure theory to homogeneous Carnot groups and the polar coordinates with horizontal radial curves constructed by Balogh and Tyson for polarizable Carnot groups. As consequences, we develop applications of potentials for sub-Laplacians in CR geometry, quaternionic CR geometry, and octonionic CR geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18221 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On potentials for sub-Laplacians and geometric applications Ma, Shiguang Qing, Jie Differential Geometry Analysis of PDEs 43A80, 35J70, 35H20, 35A08, 31C05, 31C15, 35B50, 22E60, 53C21, 31B35, 31B05, 31B15 In this paper we extend the research on potential theory and its geometric applications from Euclidean spaces to homogeneous Carnot groups. We introduce a new approach to use the geometric completeness to estimate the Hausdorff dimension of polar sets of potentials of nonnegative Radon measures for sub-Laplacians in homogeneous Carnot groups. Our approach relies on inequalities that are analogous to the classic integral inequalities about Riesz potentials in Euclidean spaces. Our approach also uses extensions of some of geometric measure theory to homogeneous Carnot groups and the polar coordinates with horizontal radial curves constructed by Balogh and Tyson for polarizable Carnot groups. As consequences, we develop applications of potentials for sub-Laplacians in CR geometry, quaternionic CR geometry, and octonionic CR geometry. |
| title | On potentials for sub-Laplacians and geometric applications |
| topic | Differential Geometry Analysis of PDEs 43A80, 35J70, 35H20, 35A08, 31C05, 31C15, 35B50, 22E60, 53C21, 31B35, 31B05, 31B15 |
| url | https://arxiv.org/abs/2512.18221 |