Proposal: DMS-9970469 Principal Investigator: Christoph M. Thiele
Abstract: The purpose of this project is to develop further the theory of multilinear singular integrals. This research started with the discovery by Michael Lacey and the Principal Investigator of the boundedness of the bilinear Hilbert transform on various L^p spaces. Their approach to this work, which answered a long-standing question of Alberto Calderon, exploited methods of time-frequency analysis similar to those found in Lennart Carleson's and Charles Fefferman's proofs of almost everywhere convergence of Fourier series. More recent results in this area include boundedness of the maximal truncated bilinear Hilbert transform, boundedness of the bisublinear maximal operator, and certain uniform estimates for the bilinear Hilbert transform, which yield a new proof for the boundedness of Calderon's first commutator. Research topics for this project include, but are by no means limited to, an extension of the aforementioned results from bilinear to multilinear singular integrals, as well as from the one-dimensional setting to higher dimensions.
This project lies in the field of analysis, which has been a central area of mathematics ever since the emergence of rigorous mathematics in the last century. More precisely, the project focuses on localized Fourier analysis, a theory that is readily described to any musician. In the same way that a musical score describes an intricate composition by breaking it down into its elementary units, notes that have specified location, duration and frequency, localized Fourier analysis synthesizes complicated mathematical functions from elementary pieces, each of which is a relatively simple function that has its characteristic time, duration, and frequency. It should come as no great surprise, therefore, that localized Fourier analysis has been applied so successfully in the automated processing of acoustic data. The specific topics in this project arose from and will have impact on the study of nonlinear problems in partial differential equations, a branch of mathematics with ever growing importance in engineering and the physical sciences. In particular, solutions to the problems under investigation could have implications for such concrete areas as data compression, signal processing, and medical imaging.
