Time frequency analysis and the bilinear Hilbert transform

The bilinear Hilbert transform is defined by It has been shown recently that this bilinear operator satisfies estimates of the type for certain p, q, and r. This is the analogue of the classical Riesz-Kolmogorow theorem for the (linear) Hilbert transform. The purpose of this talk is to discuss these and related results. A main tool is the decomposition of functions into wave packets, i.e., well localized functions whose Fourier transform is also well localized. Such decompositions are frequently used in applied mathematics such as signal or image processing.