A sharp Hörmander condition for bilinear Fourier multipliers with Lipschitz singularities

Abstract

This paper studies the $L^p$ boundedness of bilinear Fourier multipliers in the local $L^2$ 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.