Restriction estimates for hyperbolic cone in high dimensions

Gao, Zhong; Zheng, Jiqiang

doi:10.1090/proc/16259

Restriction estimates for hyperbolic cone in high dimensions
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by Zhong Gao and Jiqiang Zheng;

Proc. Amer. Math. Soc. 151 (2023), 1963-1977

DOI: https://doi.org/10.1090/proc/16259

Published electronically: February 28, 2023

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Abstract:

In this paper, we obtain the $L^p$ restriction estimates for the truncated conic surface \begin{equation*} \Sigma =\big \{(\xi ’,\xi _n,-\xi _n^{-1}\langle \xi ’,N\xi ’\rangle ): (\xi ’,\xi _n)\in B^{n-1}(0,1)\times [1,2]\big \} \end{equation*} with $N=I_{n-1-m}\oplus (-I_m)$ for $m\leq \lfloor \tfrac {n-3}2\rfloor$ provided $p>\tfrac {2(n+3)}{n+1}$. The main ingredients of the proof are the bilinear estimates of strongly separated property and a geometric distribution about caps.

References

B. Barcelo, On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc., 292(1)(1985), 321-333.
Alex Barron, Restriction estimates for hyperbolic paraboloids in higher dimensions via bilinear estimates, Rev. Mat. Iberoam. 38 (2022), no. 5, 1453–1471. MR 4502071, DOI 10.4171/rmi/1310
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
Jean Bourgain and Larry Guth, Bounds on oscillatory integral operators based on multilinear estimates, Geom. Funct. Anal. 21 (2011), no. 6, 1239–1295. MR 2860188, DOI 10.1007/s00039-011-0140-9
Benjamin Baker Bruce, Fourier restriction to a hyperbolic cone, J. Funct. Anal. 279 (2020), no. 3, 108554, 16. MR 4093785, DOI 10.1016/j.jfa.2020.108554
Chu-Hee Cho and Jungjin Lee, Improved restriction estimate for hyperbolic surfaces in $\Bbb {R}^3$, J. Funct. Anal. 273 (2017), no. 3, 917–945. MR 3653943, DOI 10.1016/j.jfa.2017.04.015
C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 284802, DOI 10.2307/2373450
C. Gao, B. Liu, C. Miao, and Y. Xi, Improved local smoothing estimate for the wave equation in higher dimensions, Preprint, arXiv:2108.06870, 2021.
Larry Guth, A restriction estimate using polynomial partitioning, J. Amer. Math. Soc. 29 (2016), no. 2, 371–413. MR 3454378, DOI 10.1090/jams827
Larry Guth, Restriction estimates using polynomial partitioning II, Acta Math. 221 (2018), no. 1, 81–142. MR 3877019, DOI 10.4310/ACTA.2018.v221.n1.a3
Sanghyuk Lee, Bilinear restriction estimates for surfaces with curvatures of different signs, Trans. Amer. Math. Soc. 358 (2006), no. 8, 3511–3533. MR 2218987, DOI 10.1090/S0002-9947-05-03796-7
Yumeng Ou and Hong Wang, A cone restriction estimate using polynomial partitioning, J. Eur. Math. Soc. (JEMS) 24 (2022), no. 10, 3557–3595. MR 4432906, DOI 10.4171/jems/1168
E. M. Stein, Some problems in harmonic analysis, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part 1, Amer. Math. Soc., Providence, RI, 1979, pp. 3–20. MR 545235
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), no. 6, 1359–1384. MR 2033842, DOI 10.1007/s00039-003-0449-0
Terence Tao, The Bochner-Riesz conjecture implies the restriction conjecture, Duke Math. J. 96 (1999), no. 2, 363–375. MR 1666558, DOI 10.1215/S0012-7094-99-09610-2
Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. MR 358216, DOI 10.1090/S0002-9904-1975-13790-6
Ana Vargas, Restriction theorems for a surface with negative curvature, Math. Z. 249 (2005), no. 1, 97–111. MR 2106972, DOI 10.1007/s00209-004-0691-7
Hong Wang, A restriction estimate in $\Bbb R^3$ using brooms, Duke Math. J. 171 (2022), no. 8, 1749–1822. MR 4484215, DOI 10.1215/00127094-2021-0064
Thomas Wolff, A sharp bilinear cone restriction estimate, Ann. of Math. (2) 153 (2001), no. 3, 661–698. MR 1836285, DOI 10.2307/2661365
A. Zygmund, On Fourier coefficients and transforms of functions of two variables, Studia Math. 50 (1974), 189–201. MR 387950, DOI 10.4064/sm-50-2-189-201

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Restriction estimates for hyperbolic cone in high dimensions
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Abstract: