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arXiv · 2402.15476

The dichotomy of Nikodym sets and local smoothing estimates for wave equations

Mingfeng Chen·Shaoming Guo
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Abstract

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.

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Mingfeng Chen, Shaoming Guo. 2024-02-23. The dichotomy of Nikodym sets and local smoothing estimates for wave equations. https://arxiv.org/abs/2402.15476

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