Sharp microlocal Kakeya--Nikodym estimates for eigenfunctions with applications
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914420782792704 |
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| author | Gao, Chuanwei Wu, Shukun Xi, Yakun |
| author_facet | Gao, Chuanwei Wu, Shukun Xi, Yakun |
| contents | We extend the microlocal Kakeya--Nikodym bounds for eigenfunctions of Blair--Sogge to a larger range of exponents, which is optimal in all dimensions $n\ge3$ on general manifolds. On manifolds of constant sectional curvature, we introduce a new anisotropic variant of the microlocal Kakeya--Nikodym norm that further enlarges the admissible $p$-range. As a corollary, by combining our results with a recent theorem of Hou, we obtain improved $L^p$ bounds for Hecke--Maass forms on compact hyperbolic $3$-manifolds.
In particular, our method applies to general Hörmander operators, and we characterize the $L^q \to L^p$ boundedness of Hörmander operators with positive-definite phase in all dimensions $n\ge3$, thereby fully resolving a question going back to Hörmander. Further applications include improved $L^q \to L^p$ Fourier extension bounds, and improved bounds related to the Bochner--Riesz conjecture in $\mathbb R^3$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01116 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp microlocal Kakeya--Nikodym estimates for eigenfunctions with applications Gao, Chuanwei Wu, Shukun Xi, Yakun Classical Analysis and ODEs Analysis of PDEs Spectral Theory We extend the microlocal Kakeya--Nikodym bounds for eigenfunctions of Blair--Sogge to a larger range of exponents, which is optimal in all dimensions $n\ge3$ on general manifolds. On manifolds of constant sectional curvature, we introduce a new anisotropic variant of the microlocal Kakeya--Nikodym norm that further enlarges the admissible $p$-range. As a corollary, by combining our results with a recent theorem of Hou, we obtain improved $L^p$ bounds for Hecke--Maass forms on compact hyperbolic $3$-manifolds. In particular, our method applies to general Hörmander operators, and we characterize the $L^q \to L^p$ boundedness of Hörmander operators with positive-definite phase in all dimensions $n\ge3$, thereby fully resolving a question going back to Hörmander. Further applications include improved $L^q \to L^p$ Fourier extension bounds, and improved bounds related to the Bochner--Riesz conjecture in $\mathbb R^3$. |
| title | Sharp microlocal Kakeya--Nikodym estimates for eigenfunctions with applications |
| topic | Classical Analysis and ODEs Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2509.01116 |