Fourier Analysis and Multilinear Operators

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The study of multilinear operators has developed largely by the interest in very concrete and naturally appearing examples whose understanding has produced powerful time-frequency tools. The investigator will capitalize on the recent successes and progresses achieved and confront problems that are needed to complete the development of the subject. At the same time novel problems and innovative approaches to study them will be introduced and explored. Singular integrals have played a crucial role in questions related to elliptic regularity, generalized Cauchy-Riemann equations, Sobolev embeddings, Littlewood-Paley theory, and other problems in partial differential equations and the study of function spaces. Multilinear singular integrals represent a next step in the proven successful enterprise of computing operations on functions via their spectral resolution and the understanding of them through the revealing lenses of the Fourier transform. Among the specific goals of the proposal are the study of bilinear or multilinear operators with minimal regularity assumptions; the development of more precise extrapolation and weighted estimates techniques; and the analysis of bilinear multipliers with singularities in the frequency domain not well-understood yet, but which are of critical importance for further progress in the field. Recent work on multilinear operators has already found both foreseen as well as unexpected applications outside harmonic analysis. Some of the questions to be investigated in this proposal can have potential impact in other mathematical disciplines, in particular in partial differential equations. Fourier analysis methods provide ways to analyze information by decomposing it into simple building blocks or wavelike components. Particular operations are easy to perform in these components by exploiting the fact that waves which oscillate at different frequencies do not interact much with each other. This is a basic idea in the analysis of signals and their transformations, and progress in the development of new Fourier analysis techniques often translates into schemes and algorithms for compression of information, pattern recognition, and other application of image processing in science and engineering. The investigator will continue to interact with graduate students and colleagues in the early stages of their careers. His research will be integrated with his teaching, training, and mentoring activities, which include the direction of PhD students working under his supervision. He will also continue to conduct research opportunities for undergraduates and participate as faculty mentor in existing programs to increase student diversity at his institution. The research in this proposal will be disseminated both through professional conferences in the discipline as well as expository lectures to broader audiences, which are intended to increase the awareness about mathematics and science in the general public.