On curvature and the bilinear multiplier problem

We provide sufficient normal curvature conditions on the boundary of a domain $D \subset \BBR^4$ to guarantee unboundedness of the bilinear Fourier multiplier operator $\T_D$ with symbol $\chi_D$ outside the local $L^2$ setting, \textit{i.e}. from $L^{p_1} (\BBR^2) \times L^{p_2} (\BBR^2) \to L^{p_3'} (\BBR^2)$ with $\sum \frac{1}{p_j} = 1$ and $p_j <2$ for some $j$. In particular, these curvature conditions are satisfied by any domain $D$ that is locally strictly convex at a single boundary point.