This paper studies the $L^{ p}$ boundedness of bilinear Fourier multipliers in the local $L^{ 2}$ range. We assume a H\"{ o}rmander condition relative to a singular set that is a finite union of Lipschitz curves. The H\"{ o}rmander 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.
Categories: math.CA