On curvature and the bilinear multiplier problem

Abstract

We provide sufficient normal curvature conditions on the boundary of a domain $D\subset {\mathbb{R}}^4$ to guarantee unboundedness of the bilinear Fourier multiplier operator ${\mathrm{T}}_D$ with symbol ${\chi }_D$ outside the local $L^2$ setting, i.e., from $L^{p_1}({\mathbb{R}}^2)\times L^{p_2}({\mathbb{R}}^2)\to L^{p_3^{\prime }}({\mathbb{R}}^2)$ 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.