Boundedness of bilinear multipliers whose symbols have a narrow support

This work is devoted to studying the boundedness on Lebesgue spaces of bilinear multipliers on $\R$ whose symbol is narrowly supported around a curve (in the frequency plane). We are looking for the optimal decay rate (depending on the width of this support) for exponents satisfying a sub-H\"older scaling. As expected, the geometry of the curve plays an important role, which is described. This has applications for the bilinear Bochner-Riesz problem (in particular, boundedness of multipliers whose symbol is the characteristic function of a set), as well as for the bilinear restriction-extension problem.