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On Kakeya-Nikodym type maximal inequalities


Author: Yakun Xi
Journal: Trans. Amer. Math. Soc. 369 (2017), 6351-6372
MSC (2010): Primary 42B25
DOI: https://doi.org/10.1090/tran/6846
Published electronically: March 31, 2017
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Abstract: We show that for any dimension $ d\ge 3$, one can obtain Wolff's $ L^{(d+2)/2}$ bound on Kakeya-Nikodym maximal function in $ \mathbb{R}^d$ for $ d\ge 3$ without the induction on scales argument. The key ingredient is to reduce to a 2-dimensional $ L^2$ estimate with an auxiliary maximal function. We also prove that the same $ L^{(d+2)/2}$ bound holds for Nikodym maximal function for any manifold $ (M^d,g)$ with constant curvature, which generalizes Sogge's results for $ d=3$ to any $ d\ge 3$. As in the 3-dimensional case, we can handle manifolds of constant curvature due to the fact that, in this case, two intersecting geodesics uniquely determine a 2-dimensional totally geodesic submanifold, which allows the use of the auxiliary maximal function.


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Additional Information

Yakun Xi
Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
Email: ykxi@math.jhu.edu

DOI: https://doi.org/10.1090/tran/6846
Keywords: Kakeya maximal function, Nikodym maximal function, Geometric combinatorics
Received by editor(s): May 20, 2015
Received by editor(s) in revised form: September 23, 2015
Published electronically: March 31, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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