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arXiv · 2210.08320

Nikodym sets and maximal functions associated with spheres

Alan Chang·Georgios Dosidis·Jongchon Kim
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Abstract

We study spherical analogues of Nikodym sets and related maximal functions. In particular, we prove sharp $L^p$-estimates for Nikodym maximal functions associated with spheres. As a corollary, any Nikodym set for spheres must have full Hausdorff dimension. In addition, we consider a class of maximal functions which contains the spherical maximal function as a special case. We show that $L^p$-estimates for these maximal functions can be deduced from local smoothing estimates for the wave equation relative to fractal measures.

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Alan Chang, Georgios Dosidis, Jongchon Kim. 2022-10-15. Nikodym sets and maximal functions associated with spheres. https://doi.org/10.4171/rmi/1519

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