On Kakeya-Nikodym type maximal inequalities
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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.References
- A. S. Besicovitch, On Kakeya’s problem and a similar one, Math. Z. 27 (1928), no. 1, 312–320. MR 1544912, DOI 10.1007/BF01171101
- J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. MR 1097257, DOI 10.1007/BF01896376
- J. Bourgain, On the dimension of Kakeya sets and related maximal inequalities, Geom. Funct. Anal. 9 (1999), no. 2, 256–282. MR 1692486, DOI 10.1007/s000390050087
- Antonio Cordoba, The Kakeya maximal function and the spherical summation multipliers, Amer. J. Math. 99 (1977), no. 1, 1–22. MR 447949, DOI 10.2307/2374006
- S. W. Drury, $L^{p}$ estimates for the X-ray transform, Illinois J. Math. 27 (1983), no. 1, 125–129. MR 684547, DOI 10.1215/ijm/1256065417
- S. Kakeya, Some problems on maximum and minimum regarding ovals, Tôhoku Science Reports 6 (1917), 71–88.
- Nets Hawk Katz, Izabella Łaba, and Terence Tao, An improved bound on the Minkowski dimension of Besicovitch sets in $\textbf {R}^3$, Ann. of Math. (2) 152 (2000), no. 2, 383–446. MR 1804528, DOI 10.2307/2661389
- Nets Hawk Katz and Terence Tao, New bounds for Kakeya problems, J. Anal. Math. 87 (2002), 231–263. Dedicated to the memory of Thomas H. Wolff. MR 1945284, DOI 10.1007/BF02868476
- Changxing Miao, Jianwei Yang, and Jiqiang Zheng, On Wolff’s $L^{\frac 52}$-Kakeya maximal inequality in $\Bbb {R}^3$, Forum Math. 27 (2015), no. 5, 3053–3077. MR 3393389, DOI 10.1515/forum-2013-0160
- William P. Minicozzi II and Christopher D. Sogge, Negative results for Nikodym maximal functions and related oscillatory integrals in curved space, Math. Res. Lett. 4 (1997), no. 2-3, 221–237. MR 1453056, DOI 10.4310/MRL.1997.v4.n2.a5
- Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), no. 1, 65–130. MR 1168960, DOI 10.1090/S0894-0347-1993-1168960-6
- Christopher D. Sogge, Concerning Nikodým-type sets in $3$-dimensional curved spaces, J. Amer. Math. Soc. 12 (1999), no. 1, 1–31. MR 1639543, DOI 10.1090/S0894-0347-99-00289-1
- Thomas Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), no. 3, 651–674. MR 1363209, DOI 10.4171/RMI/188
Additional Information
- Yakun Xi
- Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland 21218
- MR Author ID: 1178425
- Email: ykxi@math.jhu.edu
- Received by editor(s): May 20, 2015
- Received by editor(s) in revised form: September 23, 2015
- Published electronically: March 31, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 6351-6372
- MSC (2010): Primary 42B25
- DOI: https://doi.org/10.1090/tran/6846
- MathSciNet review: 3660224