A sharp H\"ormander condition for bilinear Fourier multipliers with Lipschitz singularities

This paper studies the L^p boundedness of bilinear Fourier multipliers in the local L^2 range. We assume a H\"ormander condition relative to a singular set that is a finite union of Lipschitz curves. The H\"ormander condition is sharp with respect to the Sobolev exponent. Our setup generalizes the non-degenerate bilinear Hilbert transform but avoids issues of uniform bounds near degeneracy.