A convolution estimate for two-dimensional hypersurfaces

TL;DR

This paper generalizes a convolution estimate for three hypersurfaces in R^3, extending previous results to less regular surfaces and providing uniform L^2 bounds with applications to nonlinear dispersive equations.

Contribution

It extends the Bennett-Carbery-Wright convolution estimate to C^{1,beta} hypersurfaces in R^3 under scalable assumptions, broadening the class of surfaces for which the estimate holds.

Findings

Established a uniform L^2 estimate for convolutions on less regular hypersurfaces.

Extended the nonlinear Loomis-Whitney inequality to C^{1,beta} surfaces.

Applied the results to nonlinear dispersive equations.

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.