Forms associated with the superposition of bilinear Hilbert transforms appear in many contexts in analysis. For example, developing calculus for pseudo differential operators and studying Cauchy intergrals on Lipschitz curves.

In view of these applications, the question of uniform bounds for such bilinear Hilbert transforms arose. We will explore this problem with a special focus on the multidimensional case. In particular, we will describe the main tool in the time-frequency analysis of such operators, the phase plane projection. This projection concerns the appropriate simultaneous localization of both a function and its Fourier transform to specific regions of the time-frequency support.

This talk is based on joint work with Olli Saari, Christoph Thiele, and Gennady Uraltsev.