A convolution estimate for two-dimensional hypersurfaces

Abstract

Given three transversal and sufficiently regular hypersurfaces in Rdbl;3 it follows from work of Bennett-Carbery-Wright that the convolution of two L2 functions supported of the first and second hypersurface, respectively, can be restricted to an L2 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'β hypersurfaces in Rdbl;3, under scaleable assumptions. The resulting uniform L2 estimate has applications to nonlinear dispersive equations.