Multilinearity in one and two dimensions

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This proposal describes research plans that are aimed at answering an array of questions in ergodic theory, harmonic analysis and arithmetic combinatorics that are of great interest in modern analysis. The main common feature of all these problems resides in their multilinear nature. Of particular interest when dealing with a multilinear operator acting on some product of Lebesgue spaces is to understand the range of indices for which it is well behaved. In this proposal the focus is mainly on almost everywhere convergence, and thus inherently on the boundedness of the associated maximal operators. A famous unresolved problem regards pointwise convergence for the bilinear ergodic averages associated with commuting transformations. The analysis of this question revealed deep connections with a two dimensional version of the bilinear Hilbert transform and with Carleson's theorem on the pointwise convergence of Fourier series. Another interesting circle of questions arises in the analysis of bilinear polynomial averages. Recent progress from arithmetic combinatorics, combined with multi-scale time-frequency techniques is likely to shed light on all these issues.This proposed research is at the cutting edge of what is now being done in dynamical systems, harmonic analysis and arithmetic combinatorics. It is expected that the resolution of the questions advanced in this proposal will further the mathematical community's understanding of the connections between these areas, in particular between processes in harmonic analysis and their analogues in ergodic theory. The nature of this research makes it also likely for our investigation to shed yet more light on some of the tools that are used in other areas of science, such as signal processing. The research project that is proposed in this grant will lead to interactions between the PI and mathematicians from other universities with whom part of the investigation might be conducted.