Nikodym sets and maximal functions associated with spheres
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.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2022
- DOI:
- 10.48550/arXiv.2210.08320
- arXiv:
- arXiv:2210.08320
- Bibcode:
- 2022arXiv221008320C
- Keywords:
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- Mathematics - Classical Analysis and ODEs
- E-Print:
- 37 pages, 2 figures