The dichotomy of Nikodym sets and local smoothing estimates for wave equations
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914689828519936 |
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| author | Chen, Mingfeng Guo, Shaoming |
| author_facet | Chen, Mingfeng Guo, Shaoming |
| contents | We show that Nikodym sets and local smoothing estimates for linear wave equations form a dichotomy: If Nikodym sets for a family of curves exist, then the related maximal operator is not bounded on $L^p(\mathbb{R}^2)$ for any $p<\infty$; if Nikodym sets do not exist, then local smoothing estimates hold, and the related maximal operator is bounded on $L^p(\mathbb{R}^2)$ for some $p<\infty$. Whenever the maximal operator is bounded on $L^p(\mathbb{R}^2)$ for some $p<\infty$, we also determine the sharp exponent for $L^p(\mathbb{R}^2)$ bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_15476 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The dichotomy of Nikodym sets and local smoothing estimates for wave equations Chen, Mingfeng Guo, Shaoming Classical Analysis and ODEs We show that Nikodym sets and local smoothing estimates for linear wave equations form a dichotomy: If Nikodym sets for a family of curves exist, then the related maximal operator is not bounded on $L^p(\mathbb{R}^2)$ for any $p<\infty$; if Nikodym sets do not exist, then local smoothing estimates hold, and the related maximal operator is bounded on $L^p(\mathbb{R}^2)$ for some $p<\infty$. Whenever the maximal operator is bounded on $L^p(\mathbb{R}^2)$ for some $p<\infty$, we also determine the sharp exponent for $L^p(\mathbb{R}^2)$ bounds. |
| title | The dichotomy of Nikodym sets and local smoothing estimates for wave equations |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2402.15476 |