Bourgain's condition, sticky Kakeya, and new examples
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917398794207232 |
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| author | Nadjimzadah, Arian |
| author_facet | Nadjimzadah, Arian |
| contents | We prove that in all dimensions at least 3 and for any Hörmander-type oscillatory integral operator satisfying Bourgain's condition, the sticky case of the corresponding curved Kakeya conjecture reduces to the sticky case of the classical Kakeya conjecture. This supports a conjecture of Guo-Wang-Zhang, that an operator satisfies the same $L^p$ bounds as in the restriction conjecture exactly when it satisfies Bourgain's condition.
Our result follows from a new geometric characterization of Bourgain's condition based on the structure of curved $δ$-tubes in a $δ^{1/2}$-tube. We find examples in all dimensions at least 3 which show this property does not persist in a larger tube, and in particular these are the first operators satisfying Bourgain's condition for which there is no diffeomorphism taking the corresponding families of curves to lines. This suggests that a general to sticky reduction in the spirit of Wang-Zahl needs substantial new ideas. We expect these examples to provide a good starting point. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_10918 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bourgain's condition, sticky Kakeya, and new examples Nadjimzadah, Arian Classical Analysis and ODEs We prove that in all dimensions at least 3 and for any Hörmander-type oscillatory integral operator satisfying Bourgain's condition, the sticky case of the corresponding curved Kakeya conjecture reduces to the sticky case of the classical Kakeya conjecture. This supports a conjecture of Guo-Wang-Zhang, that an operator satisfies the same $L^p$ bounds as in the restriction conjecture exactly when it satisfies Bourgain's condition. Our result follows from a new geometric characterization of Bourgain's condition based on the structure of curved $δ$-tubes in a $δ^{1/2}$-tube. We find examples in all dimensions at least 3 which show this property does not persist in a larger tube, and in particular these are the first operators satisfying Bourgain's condition for which there is no diffeomorphism taking the corresponding families of curves to lines. This suggests that a general to sticky reduction in the spirit of Wang-Zahl needs substantial new ideas. We expect these examples to provide a good starting point. |
| title | Bourgain's condition, sticky Kakeya, and new examples |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2511.10918 |