Spectral multipliers on two-step stratified Lie groups with degenerate group structure
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916605572677632 |
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| author | Niedorf, Lars |
| author_facet | Niedorf, Lars |
| contents | Let $L$ be a sub-Laplacian on a two-step stratified Lie group $G$ of topological dimension $d$. We prove new $L^p$-spectral multiplier estimates under the sharp regularity condition $s>d\left|1/p-1/2\right|$ in settings where the group structure of $G$ is degenerate, extending previously known results for the non-degenerate case. Our results include variants of the free two-step nilpotent group on three generators and Heisenberg-Reiter groups. The proof combines restriction type estimates with a detailed analysis of the sub-Riemannian geometry of $G$. A key novelty of our approach is the use of a refined spectral decomposition into caps on the unit sphere in the center of the Lie group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16262 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral multipliers on two-step stratified Lie groups with degenerate group structure Niedorf, Lars Analysis of PDEs 42B15, 22E25, 22E30, 43A85 Let $L$ be a sub-Laplacian on a two-step stratified Lie group $G$ of topological dimension $d$. We prove new $L^p$-spectral multiplier estimates under the sharp regularity condition $s>d\left|1/p-1/2\right|$ in settings where the group structure of $G$ is degenerate, extending previously known results for the non-degenerate case. Our results include variants of the free two-step nilpotent group on three generators and Heisenberg-Reiter groups. The proof combines restriction type estimates with a detailed analysis of the sub-Riemannian geometry of $G$. A key novelty of our approach is the use of a refined spectral decomposition into caps on the unit sphere in the center of the Lie group. |
| title | Spectral multipliers on two-step stratified Lie groups with degenerate group structure |
| topic | Analysis of PDEs 42B15, 22E25, 22E30, 43A85 |
| url | https://arxiv.org/abs/2501.16262 |