A note on bilinear multipliers with convex singularities

We study bounds in the local $L^2$ range of exponents for bilinear multipliers whose symbol is the characteristic function of the epigraph of certain convex curves. We realize these bounds as a consequence of estimates that we establish, via simple arguments, for the associated exotic paraproducts. As a further application, we observe bounds beyond the local $L^2$ range for bilinear multipliers whose symbol is the characteristic function of the epigraph of convex polygonal curves associated with these paraproducts.