A sharp Hörmander condition for bilinear Fourier multipliers with Lipschitz singularities

Abstract This paper studies the $$L^{p}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> boundedness of bilinear Fourier multipliers in the local $$L^{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> range. We assume a Hörmander condition relative to a singular set that is a finite union of Lipschitz curves. The Hörmander 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.