Revista Matemática Iberoamericana

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Volume 26, Issue 2, 2010, pp. 707–728
DOI: 10.4171/RMI/615

Published online: 2010-08-31

A convolution estimate for two-dimensional hypersurfaces

Ioan Bejenaru[1], Sebastian Herr[2] and Daniel Tataru[3]

(1) University of Chicago, United States
(2) Universität Bielefeld, Germany
(3) University of California, Berkeley, USA

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.

Keywords: Transversality, hypersurface, convolution, $L^2$ estimate, induction on scales

Bejenaru Ioan, Herr Sebastian, Tataru Daniel: A convolution estimate for two-dimensional hypersurfaces. Rev. Mat. Iberoamericana 26 (2010), 707-728. doi: 10.4171/RMI/615