Failure of stability of a maximal operator bound for perturbed Nevo-Thangavelu means
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910109836247040 |
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| author | Ryu, Jaehyeon Seeger, Andreas |
| author_facet | Ryu, Jaehyeon Seeger, Andreas |
| contents | Let $G$ be a two-step nilpotent Lie group, identified via the exponential map with the Lie-algebra $\mathfrak g=\mathfrak g_1\oplus\mathfrak g_2$, where $[\mathfrak g,\mathfrak g]\subset \mathfrak g_2$. We consider maximal functions associated to spheres in a $d$-dimensional linear subspace $H$, dilated by the automorphic dilations. $L^p$ boundedness results for the case where $H=\mathfrak g_1$ are well understood. Here we consider the case of a tilted hyperplane $H\neq \mathfrak g_1$ which is not invariant under the automorphic dilations. In the case of Métivier groups it is known that the $L^p$-boundedness results are stable under a small linear tilt. We show that this is generally not the case for other two-step groups, and provide new necessary conditions for $L^p$ boundedness. We prove these results in a more general setting with tilted versions of submanifolds of $\mathfrak g_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_14180 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Failure of stability of a maximal operator bound for perturbed Nevo-Thangavelu means Ryu, Jaehyeon Seeger, Andreas Classical Analysis and ODEs Functional Analysis Representation Theory Let $G$ be a two-step nilpotent Lie group, identified via the exponential map with the Lie-algebra $\mathfrak g=\mathfrak g_1\oplus\mathfrak g_2$, where $[\mathfrak g,\mathfrak g]\subset \mathfrak g_2$. We consider maximal functions associated to spheres in a $d$-dimensional linear subspace $H$, dilated by the automorphic dilations. $L^p$ boundedness results for the case where $H=\mathfrak g_1$ are well understood. Here we consider the case of a tilted hyperplane $H\neq \mathfrak g_1$ which is not invariant under the automorphic dilations. In the case of Métivier groups it is known that the $L^p$-boundedness results are stable under a small linear tilt. We show that this is generally not the case for other two-step groups, and provide new necessary conditions for $L^p$ boundedness. We prove these results in a more general setting with tilted versions of submanifolds of $\mathfrak g_1$. |
| title | Failure of stability of a maximal operator bound for perturbed Nevo-Thangavelu means |
| topic | Classical Analysis and ODEs Functional Analysis Representation Theory |
| url | https://arxiv.org/abs/2510.14180 |