JournalsrmiVol. 26, No. 2pp. 707–728

A convolution estimate for two-dimensional hypersurfaces

  • Ioan Bejenaru

    University of Chicago, United States
  • Sebastian Herr

    Universität Bielefeld, Germany
  • Daniel Tataru

    University of California, Berkeley, USA
A convolution estimate for two-dimensional hypersurfaces cover
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Abstract

Given three transversal and sufficiently regular hypersurfaces in R3\mathbb{R}^3 it follows from work of Bennett-Carbery-Wright that the convolution of two L2L^2 functions supported of the first and second hypersurface, respectively, can be restricted to an L2L^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 C1,βC^{1,\beta} hypersurfaces in R3\mathbb{R}^3, under scaleable assumptions. The resulting uniform L2L^2 estimate has applications to nonlinear dispersive equations.

Cite this article

Ioan Bejenaru, Sebastian Herr, Daniel Tataru, A convolution estimate for two-dimensional hypersurfaces. Rev. Mat. Iberoam. 26 (2010), no. 2, pp. 707–728

DOI 10.4171/RMI/615