A convolution estimate for two-dimensional hypersurfaces

Abstract

Given three transversal and sufficiently regular hypersurfaces in 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 R^3, under scaleable assumptions. The resulting uniform L^2 estimate has applications to nonlinear dispersive equations.