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A restriction estimate using polynomial partitioning

Author: Larry Guth
Journal: J. Amer. Math. Soc. 29 (2016), 371-413
MSC (2010): Primary 42B20
Published electronically: May 11, 2015
MathSciNet review: 3454378
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Abstract: If $ S$ is a smooth compact surface in $ \mathbb{R}^3$ with strictly positive second fundamental form, and $ E_S$ is the corresponding extension operator, then we prove that for all $ p > 3.25$, $ \Vert E_S f\Vert _{L^p(\mathbb{R}^3)} \le C(p,S) \Vert f \Vert _{L^\infty (S)}$. The proof uses polynomial partitioning arguments from incidence geometry.

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  • [B] Jean Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147-187. MR 1097257 (92g:42010),
  • [BG] 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 (2012k:42018),
  • [BCT] Jonathan Bennett, Anthony Carbery, and Terence Tao, On the multilinear restriction and Kakeya conjectures, Acta Math. 196 (2006), no. 2, 261-302. MR 2275834 (2007h:42019),
  • [CKW] Xi Chen, Neeraj Kayal, and Avi Wigderson, Partial derivatives in arithmetic complexity and beyond, Found. Trends Theor. Comput. Sci. 6 (2010), no. 1-2, front matter, 1-138 (2011). MR 2901512,
  • [CEGSW] Kenneth L. Clarkson, Herbert Edelsbrunner, Leonidas J. Guibas, Micha Sharir, and Emo Welzl, Combinatorial complexity bounds for arrangements of curves and spheres, Discrete Comput. Geom. 5 (1990), no. 2, 99-160. MR 1032370 (91f:52021),
  • [C] Antonio Córdoba, Geometric Fourier analysis, Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, vii, 215-226 (English, with French summary). MR 688026 (84i:42029)
  • [D] Zeev Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (2009), no. 4, 1093-1097. MR 2525780 (2011a:52039),
  • [GP] Victor Guillemin and Alan Pollack, Differential topology, AMS Chelsea Publishing, Providence, RI, 2010. Reprint of the 1974 original. MR 2680546 (2011e:58001)
  • [G1] Larry Guth, Degree reduction and graininess for Kakeya-type sets in $ \mathbb{R}^3$, available at arXiv:1402.0518.
  • [G2] Larry Guth, Distinct distance estimates and low degree polynomial partitioning, available at arXiv:1404.2321.
  • [GK] Larry Guth and Nets Katz, On the Erdős distinct distance problem in the plane, available at arXiv:1011.4105.
  • [KMS] Haim Kaplan, Jiří Matoušek, and Micha Sharir, Simple proofs of classical theorems in discrete geometry via the Guth-Katz polynomial partitioning technique, Discrete Comput. Geom. 48 (2012), no. 3, 499-517. MR 2957631,
  • [Ma] Jiří Matoušek, Using the Borsuk-Ulam theorem, Universitext, Springer-Verlag, Berlin, 2003. Lectures on topological methods in combinatorics and geometry; Written in cooperation with Anders Björner and Günter M. Ziegler. MR 1988723 (2004i:55001)
  • [Mi] J. Milnor, On the Betti numbers of real varieties, Proc. Amer. Math. Soc. 15 (1964), 275-280. MR 0161339 (28 #4547)
  • [SS] Micha Sharir, The interface between computational and combinatorial geometry, Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2005, pp. 137-145 (electronic). MR 2298259
  • [SoTa] József Solymosi and Terence Tao, An incidence theorem in higher dimensions, Discrete Comput. Geom. 48 (2012), no. 2, 255-280. MR 2946447,
  • [St] 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, Amer. Math. Soc., Providence, R.I., 1979, pp. 3-20. MR 545235 (80m:42027)
  • [StTu] A. H. Stone and J. W. Tukey, Generalized ``sandwich'' theorems, Duke Math. J. 9 (1942), 356-359. MR 0007036 (4,75c)
  • [SzTr] Endre Szemerédi and William T. Trotter Jr., Extremal problems in discrete geometry, Combinatorica 3 (1983), no. 3-4, 381-392. MR 729791 (85j:52014),
  • [TVV] Terence Tao, Ana Vargas, and Luis Vega, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), no. 4, 967-1000. MR 1625056 (99f:42026),
  • [T1] Terence Tao, The Bochner-Riesz conjecture implies the restriction conjecture, Duke Math. J. 96 (1999), no. 2, 363-375. MR 1666558 (2000a:42023),
  • [T2] Terence Tao, A sharp bilinear restrictions estimate for paraboloids, Geom. Funct. Anal. 13 (2003), no. 6, 1359-1384. MR 2033842 (2004m:47111),
  • [T3] Terence Tao. Lecture notes on restriction, Math 254B, Spring 1999.
  • [Won] Richard Wongkew, Volumes of tubular neighbourhoods of real algebraic varieties, Pacific J. Math. 159 (1993), no. 1, 177-184. MR 1211391 (94e:14073)
  • [W1] Thomas Wolff, A sharp bilinear cone restriction estimate, Ann. of Math. (2) 153 (2001), no. 3, 661-698. MR 1836285 (2002j:42019),
  • [W2] Thomas Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), no. 3, 651-674. MR 1363209 (96m:42034),
  • [W3] Thomas Wolff, Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996) Amer. Math. Soc., Providence, RI, 1999, pp. 129-162. MR 1660476 (2000d:42010)
  • [W4] Thomas Wolff, Local smoothing type estimates on $ L^p$ for large $ p$, Geom. Funct. Anal. 10 (2000), no. 5, 1237-1288. MR 1800068 (2001k:42030),
  • [W5] Thomas Wolff, A Kakeya-type problem for circles, Amer. J. Math. 119 (1997), no. 5, 985-1026. MR 1473067 (98m:42027)
  • [Z] R. Zhang, Polynomials with dense zero sets and discrete models of the Kakeya conjecture and the Furstenberg set problem, available at arXiv:1403.1352.

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Additional Information

Larry Guth
Affiliation: Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139

Received by editor(s): July 14, 2014
Received by editor(s) in revised form: January 23, 2015
Published electronically: May 11, 2015
Additional Notes: The author is supported by a Simons Investigator Award.
Article copyright: © Copyright 2015 American Mathematical Society

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