This proposal describes research plans that are aimed at answering an array of questions in ergodic theory and harmonic analysis that are of great interest in modern analysis. The main common feature of all these problems resides in their multi-linear nature. Of particular Interest, when dealing with a multi-linear 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. An important instance of multi-linearity is represented by the combinatorial averages such as Furstenberg's nonstandard averages and the averages on cubes. The convergence of such averages has deep implications for various Ramsey-type problems on integers and even on general abstract groups. A second type of multi-linear operators we plan to investigate is represented by the weighted averages and series, such as the ones appearing in the so-called ``return times theorems''. The analysis of these objects reveals striking connections between dynamics and time-frequency analysis, in particular between Birghoff's point-wise ergodic theorem and Carleson's result on the point-wise convergence of Fourier series.
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 analogue 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.
