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.

]]>We stress that the telegraph process solves a partial linear differential equation of the hyperbolic type for which explicit computations can be carried by in terms of Bessel functions. In the present talk, I will discuss a coupling approach, which is a robust technique that in principle can be used for more general PDEs. The proof is done via the interplay of the following couplings: coin-flip coupling, synchronous coupling and the celebrated Komlós–Major–Tusnády coupling. In addition, nonasymptotic estimates for the corresponding $L^p$ time average are given explicitly.

The talk is based on joint work with Jani Lukkarinen, University of Helsinki, Finland.

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