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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On Kakeya-Nikodym type maximal inequalities

Author: Yakun Xi
Journal: Trans. Amer. Math. Soc. 369 (2017), 6351-6372
MSC (2010): Primary 42B25
Published electronically: March 31, 2017
MathSciNet review: 3660224
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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
MR Author ID: 1178425

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