Harmonic Analysis

Basic analytic questions in Nonlinear Fourier analysis

speaker: Christoph Thiele (UCLA)

abstract: Scattering transforms are often referred to as nonlinear Fourier transforms. They come in many facets, but we shall concentrate on a very simple model: consider a potential F on the real line and the corresponding Dirac operator (F-D)(F+D). Then the transmission and reflection coefficient of this operator are the nonlinear Fourioer transform of F. We are interested in nonlinear analogues of regularity estimates for the classical Fourier transform such as Plancherel identity, Hausdorff young inequality, Carleson's theorem. For example, there is a nonlinear Fourier transform for functions in the space L2(R). In contrast to the classical situation, the nonlinear Fourier transform is not injective on L2(R), and one of the questions we have is to understand the fibers of this map. The work presented is joint with T. Tao.

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