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.