Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 4 (2017), 817-891.

A Fourier restriction theorem for a two-dimensional surface of finite type

Stefan Buschenhenke, Detlef Müller, and Ana Vargas

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


The problem of Lq(3) L2(S) Fourier restriction estimates for smooth hypersurfaces S of finite type in 3 is by now very well understood for a large class of hypersurfaces, including all analytic ones. In this article, we take up the study of more general Lq(3) Lr(S) Fourier restriction estimates, by studying a prototypical model class of two-dimensional surfaces for which the Gaussian curvature degenerates in one-dimensional subsets. We obtain sharp restriction theorems in the range given by Tao in 2003 in his work on paraboloids. For high-order degeneracies this covers the full range, closing the restriction problem in Lebesgue spaces for those surfaces. A surprising new feature appears, in contrast with the nonvanishing curvature case: there is an extra necessary condition. Our approach is based on an adaptation of the bilinear method. A careful study of the dependence of the bilinear estimates on the curvature and size of the support is required.

Article information

Anal. PDE, Volume 10, Number 4 (2017), 817-891.

Received: 3 February 2016
Revised: 2 September 2016
Accepted: 22 January 2017
First available in Project Euclid: 19 October 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B10: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type

Fourier restriction finite type multilinear bilinear


Buschenhenke, Stefan; Müller, Detlef; Vargas, Ana. A Fourier restriction theorem for a two-dimensional surface of finite type. Anal. PDE 10 (2017), no. 4, 817--891. doi:10.2140/apde.2017.10.817.

Export citation


  • B. Barceló Taberner, “On the restriction of the Fourier transform to a conical surface”, Trans. Amer. Math. Soc. 292:1 (1985), 321–333.
  • B. Barceló, “The restriction of the Fourier transform to some curves and surfaces”, Studia Math. 84:1 (1986), 39–69.
  • J. Bennett, A. Carbery, and T. Tao, “On the multilinear restriction and Kakeya conjectures”, Acta Math. 196:2 (2006), 261–302.
  • J. Bourgain, “Estimations de certaines fonctions maximales”, C. R. Acad. Sci. Paris Sér. I Math. 301:10 (1985), 499–502.
  • J. Bourgain, “Besicovitch type maximal operators and applications to Fourier analysis”, Geom. Funct. Anal. 1:2 (1991), 147–187.
  • J. Bourgain, “Estimates for cone multipliers”, pp. 41–60 in Geometric aspects of functional analysis (Israel, 1992–1994), edited by J. Lindenstrauss and V. Milman, Oper. Theory Adv. Appl. 77, Birkhäuser, Basel, 1995.
  • J. Bourgain, “Some new estimates on oscillatory integrals”, pp. 83–112 in Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), edited by C. Fefferman et al., Princeton Math. Ser. 42, Princeton Univ. Press, 1995.
  • J. Bourgain and L. Guth, “Bounds on oscillatory integral operators based on multilinear estimates”, Geom. Funct. Anal. 21:6 (2011), 1239–1295.
  • S. Buschenhenke, “A sharp $L^p$-$L^q$-Fourier restriction theorem for a conical surface of finite type”, Math. Z. 280:1 (2015), 367–399.
  • C. Fefferman, “Inequalities for strongly singular convolution operators”, Acta Math. 124:1 (1970), 9–36.
  • E. Ferreyra and M. Urciuolo, “Fourier restriction estimates to mixed homogeneous surfaces”, J. Inequal. Pure Appl. Math. 10:2 (2009), art. id. 35, 11 pp.
  • L. Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics 249, Springer, 2008.
  • A. Greenleaf, “Principal curvature and harmonic analysis”, Indiana Univ. Math. J. 30:4 (1981), 519–537.
  • I. A. Ikomov and D. Müller, “$L^p-L^2$ Fourier restriction for hypersurfaces in $\mathbb R^3$, I”, preprint, 2012.
  • I. A. Ikomov and D. Müller, “$L^p-L^2$ Fourier restriction for hypersurfaces in $\mathbb R^3$, II”, preprint, 2014.
  • I. A. Ikromov and D. Müller, “Uniform estimates for the Fourier transform of surface carried measures in $\mathbb R^3$ and an application to Fourier restriction”, J. Fourier Anal. Appl. 17:6 (2011), 1292–1332.
  • I. A. Ikromov, M. Kempe, and D. Müller, “Estimates for maximal functions associated with hypersurfaces in $\mathbb R^3$ and related problems of harmonic analysis”, Acta Math. 204:2 (2010), 151–271.
  • S. Lee, “Endpoint estimates for the circular maximal function”, Proc. Amer. Math. Soc. 131:5 (2003), 1433–1442.
  • S. Lee, “Bilinear restriction estimates for surfaces with curvatures of different signs”, Trans. Amer. Math. Soc. 358:8 (2006), 3511–3533.
  • S. Lee and A. Vargas, “Restriction estimates for some surfaces with vanishing curvatures”, J. Funct. Anal. 258:9 (2010), 2884–2909.
  • A. Moyua, A. Vargas, and L. Vega, “Schrödinger maximal function and restriction properties of the Fourier transform”, Internat. Math. Res. Notices 1996:16 (1996), 793–815.
  • A. Moyua, A. Vargas, and L. Vega, “Restriction theorems and maximal operators related to oscillatory integrals in $\mathbb{R}^3$”, Duke Math. J. 96:3 (1999), 547–574.
  • J. Ramos, “The trilinear restriction estimate with sharp dependence on the transversality”, preprint, 2016.
  • C. D. Sogge, “A sharp restriction theorem for degenerate curves in ${\mathbb{R}}^2$”, Amer. J. Math. 109:2 (1987), 223–228.
  • K. Spindler, “A short proof of the formula of Faà di Bruno”, Elem. Math. 60:1 (2005), 33–35.
  • E. M. Stein, “Oscillatory integrals in Fourier analysis”, pp. 307–355 in Beijing lectures in harmonic analysis (Beijing, 1984), edited by E. M. Stein, Ann. of Math. Stud. 112, Princeton Univ. Press, 1986.
  • E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series 43, Princeton University Press, 1993.
  • B. Stovall, “Linear and bilinear restriction to certain rotationally symmetric hypersurfaces”, preprint, 2015.
  • R. S. Strichartz, “Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations”, Duke Math. J. 44:3 (1977), 705–714.
  • T. Tao, “Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates”, Math. Z. 238:2 (2001), 215–268.
  • T. Tao, “From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE”, Notices Amer. Math. Soc. 48:3 (2001), 294–303.
  • T. Tao, “Recent progress on the restriction conjecture”, preprint, 2003.
  • T. Tao, “A sharp bilinear restrictions estimate for paraboloids”, Geom. Funct. Anal. 13:6 (2003), 1359–1384.
  • T. Tao and A. Vargas, “A bilinear approach to cone multipliers, I: Restriction estimates”, Geom. Funct. Anal. 10:1 (2000), 185–215.
  • T. Tao and A. Vargas, “A bilinear approach to cone multipliers, II: Applications”, Geom. Funct. Anal. 10:1 (2000), 216–258.
  • T. Tao, A. Vargas, and L. Vega, “A bilinear approach to the restriction and Kakeya conjectures”, J. Amer. Math. Soc. 11:4 (1998), 967–1000.
  • P. A. Tomas, “A restriction theorem for the Fourier transform”, Bull. Amer. Math. Soc. 81:2 (1975), 477–478.
  • A. Vargas, “Restriction theorems for a surface with negative curvature”, Math. Z. 249:1 (2005), 97–111.
  • T. Wolff, “An improved bound for Kakeya type maximal functions”, Rev. Mat. Iberoamericana 11:3 (1995), 651–674.
  • T. Wolff, “A sharp bilinear cone restriction estimate”, Ann. of Math. $(2)$ 153:3 (2001), 661–698.
  • A. Zygmund, “On Fourier coefficients and transforms of functions of two variables”, Studia Math. 50:2 (1974), 189–201.