Revista Matemática Iberoamericana

A convolution estimate for two-dimensional hypersurfaces

Ioan Bejenaru , Sebastian Herr , and Daniel Tataru

Full-text: Open access

Abstract

Given three transversal and sufficiently regular hypersurfaces in $\mathbb{R}^3$ it follows from work of Bennett-Carbery-Wright that the convolution of two $L^2$ functions supported of the first and second hypersurface, respectively, can be restricted to an $L^2$ function on the third hypersurface, which can be considered as a nonlinear version of the Loomis-Whitney inequality. We generalize this result to a class of $C^{1,\beta}$ hypersurfaces in $\mathbb{R}^3$, under scaleable assumptions. The resulting uniform $L^2$ estimate has applications to nonlinear dispersive equations.

Article information

Source
Rev. Mat. Iberoamericana, Volume 26, Number 2 (2010), 707-728.

Dates
First available in Project Euclid: 4 June 2010

Permanent link to this document
https://projecteuclid.org/euclid.rmi/1275671317

Mathematical Reviews number (MathSciNet)
MR2677013

Zentralblatt MATH identifier
1203.42033

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis
Secondary: 47B38: Operators on function spaces (general)

Keywords
transversality hypersurface convolution $L^2$ estimate induction on scales

Citation

Bejenaru, Ioan; Herr, Sebastian; Tataru, Daniel. A convolution estimate for two-dimensional hypersurfaces. Rev. Mat. Iberoamericana 26 (2010), no. 2, 707--728. https://projecteuclid.org/euclid.rmi/1275671317


Export citation

References

  • Bejenaru, I.: Quadratic nonlinear derivative Schrödinger equations. II. Trans. Amer. Math. Soc. 360 (2008), no. 11, 5925-5957.
  • Bejenaru, I. and De Silva, D.: Low regularity solutions for a 2D quadratic nonlinear Schrödinger equation. Trans. Amer. Math. Soc. 360 (2008), no. 11, 5805-5830.
  • Bejenaru, I., Herr, S., Holmer, J. and Tataru, D.: On the 2D Zakharov system with $L^2$-Schrödinger data. Nonlinearity 22 (2009), no. 5, 1063-1089.
  • Bennett, J., Carbery, A. and Tao, T.: On the multilinear restriction and Kakeya conjectures. Acta Math. 196 (2006), no. 2, 261-302.
  • Bennett, J., Carbery, A. and Wright, J.: A non-linear generalisation of the Loomis-Whitney inequality and applications. Math. Res. Lett. 12 (2005), no. 4, 443-457.
  • Colliander, J.E., Delort, J.-M., Kenig, C.E. and Staffilani, G.: Bilinear estimates and applications to 2D NLS. Trans. Amer. Math. Soc. 353 (2001), no. 8, 3307-3325 (electronic).
  • Ionescu, A.D., Kenig, C.E. and Tataru, D.: Global well-posedness of the KP-I initial-value problem in the energy space. Invent. Math. 173 (2008), no. 2, 265-304.
  • Loomis, L.H. and Whitney, H.: An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc. 55 (1949), 961-962.
  • Tao, T.: Multilinear weighted convolution of $L^2$-functions, and applications to nonlinear dispersive equations. Amer. J. Math. 123 (2001), no. 5, 839-908.