## Revista Matemática Iberoamericana

### A convolution estimate for two-dimensional hypersurfaces

#### 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

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)

#### 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

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