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-Hölder scaling. As expected, the geometry of the curve plays an important role, which is described. This has applications to the bilinear Bochner-Riesz problem (in particular, boundedness of multipliers whose symbol is the characteristic function of a set), as well as to the bilinear restriction-extension problem.
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