## A restriction estimate using polynomial partitioning

HTML articles powered by AMS MathViewer

- by Larry Guth
- J. Amer. Math. Soc.
**29**(2016), 371-413 - DOI: https://doi.org/10.1090/jams827
- Published electronically: May 11, 2015
- PDF | Request permission

## 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 $p > 3.25$, $\| E_S f\|_{L^p(\mathbb {R}^3)} \le C(p,S) \| f \|_{L^\infty (S)}$. The proof uses polynomial partitioning arguments from incidence geometry.## References

- 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 - Jonathan Bennett, Anthony Carbery, and Terence Tao,
*On the multilinear restriction and Kakeya conjectures*, Acta Math.**196**(2006), no. 2, 261–302. MR**2275834**, DOI 10.1007/s11511-006-0006-4 - 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**, DOI 10.1561/0400000043 - 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**, DOI 10.1007/BF02187783 - Antonio Córdoba,
*Geometric Fourier analysis*, Ann. Inst. Fourier (Grenoble)**32**(1982), no. 3, vii, 215–226 (English, with French summary). MR**688026** - Zeev Dvir,
*On the size of Kakeya sets in finite fields*, J. Amer. Math. Soc.**22**(2009), no. 4, 1093–1097. MR**2525780**, DOI 10.1090/S0894-0347-08-00607-3 - Victor Guillemin and Alan Pollack,
*Differential topology*, AMS Chelsea Publishing, Providence, RI, 2010. Reprint of the 1974 original. MR**2680546**, DOI 10.1090/chel/370 - Larry Guth,
*Degree reduction and graininess for Kakeya-type sets in $\mathbb {R}^3$*, available at arXiv:1402.0518. - Larry Guth,
*Distinct distance estimates and low degree polynomial partitioning*, available at arXiv:1404.2321. - Larry Guth and Nets Katz,
*On the Erdős distinct distance problem in the plane*, available at arXiv:1011.4105. - 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**, DOI 10.1007/s00454-012-9443-3 - 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** - J. Milnor,
*On the Betti numbers of real varieties*, Proc. Amer. Math. Soc.**15**(1964), 275–280. MR**161339**, DOI 10.1090/S0002-9939-1964-0161339-9 - 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. MR**2298259** - József Solymosi and Terence Tao,
*An incidence theorem in higher dimensions*, Discrete Comput. Geom.**48**(2012), no. 2, 255–280. MR**2946447**, DOI 10.1007/s00454-012-9420-x - 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** - A. H. Stone and J. W. Tukey,
*Generalized “sandwich” theorems*, Duke Math. J.**9**(1942), 356–359. MR**7036**, DOI 10.1215/S0012-7094-42-00925-6 - Endre Szemerédi and William T. Trotter Jr.,
*Extremal problems in discrete geometry*, Combinatorica**3**(1983), no. 3-4, 381–392. MR**729791**, DOI 10.1007/BF02579194 - 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**, DOI 10.1090/S0894-0347-98-00278-1 - 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 - 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. Lecture notes on restriction, Math 254B, Spring 1999.
- Richard Wongkew,
*Volumes of tubular neighbourhoods of real algebraic varieties*, Pacific J. Math.**159**(1993), no. 1, 177–184. MR**1211391**, DOI 10.2140/pjm.1993.159.177 - Thomas Wolff,
*A sharp bilinear cone restriction estimate*, Ann. of Math. (2)**153**(2001), no. 3, 661–698. MR**1836285**, DOI 10.2307/2661365 - 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 - 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** - T. Wolff,
*Local smoothing type estimates on $L^p$ for large $p$*, Geom. Funct. Anal.**10**(2000), no. 5, 1237–1288. MR**1800068**, DOI 10.1007/PL00001652 - Thomas Wolff,
*A Kakeya-type problem for circles*, Amer. J. Math.**119**(1997), no. 5, 985–1026. MR**1473067**, DOI 10.1353/ajm.1997.0034 - R. Zhang,
*Polynomials with dense zero sets and discrete models of the Kakeya conjecture and the Furstenberg set problem*, available at arXiv:1403.1352.

## Bibliographic Information

**Larry Guth**- Affiliation: Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
- MR Author ID: 786046
- Email: lguth@math.mit.edu
- 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.
- © Copyright 2015 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**29**(2016), 371-413 - MSC (2010): Primary 42B20
- DOI: https://doi.org/10.1090/jams827
- MathSciNet review: 3454378