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.