An extension of bilinear Hilbert transform to distributions

Abstract

The bilinear Hilbert transform Hx: L2× L∞ → L2 respectively, Hx: Lp1 × Lp2 → Lp, is extended to D'L2 × DLx → D'L2, respectively, D'Lq × DLp2 → D'q1, (with suitable parameters) as a hypocontinuous, respectively, continuous mapping. The bilinear Hilbert transform of Schwartz distributions is defined as an irregular operation. The inversion formula is given. It is used as a product formula for appropriate pairs of distributions.