Fourier Analysis: Old Themes, New Perspectives

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DMS 0400387PI: Loukas GrafakosUnviersity of Missouri Fourier Analysis: Old themes, new perspectiveAbstractMany problems in Fourier Analysis are as old as the subject itself.The quest to understand some of these problems has pushed the research to such extraordinary developmentsthat new connections with many other scientific areashave been discovered andnew perspectives revealed. One of these new perspectivesis the multilinear point of view that has significantly refined and enriched our classical approach of studying problems via the method of freezing variables. The basic idea in multilinear analysis is that linearly independent functional parameters are treated as linear variables, allowing a higher degree of freedom and greater flexibility, which often results in a deeper and more far-reaching understanding.The principal investigator proposes to embark on an extensive study of translation-invariant multilinear operators, both broad enough to cover most known examples, but also deep enough to include special operators of a very singular nature. This proposal consists of three parts: positive theory, negative results, and general theory. Proposed work on the positive theory includes extension of the range of boundedness of the bilinear Hilbert transform and other multilinear rough singular integrals. Negative results focus on the disc multiplier outside the locally square integrable case, the bilinear Hilbert transform on products of integrable functions, as well as the Carleson-Hunt operator on spaces of nearly integrable functions and certain higher dimensional versions of all the previous operators.The proposed general theory will focus on topics concerningmultilinear interpolation and extrapolation.These items provide important tools that simplify the study of the subject.The main goal of Fourier Analysis is to study complicated objects via decompositions into simpler pieces. Just as music can be disassembled into compositions of integral multiples of a simple basic frequency, complicated operators can be represented by their actions on a spectrum of frequencies. Fourier analysis provides the tools that relate special (time) and phasial (frequency) aspects of the same function (signal).Irregularities of signals and images are better located once these are decomposed into small pieces and studied via Fourier analysis. For instance, noise and blurring are easily located with the application of the Fourier transform, but nowadays even more challenging feats can be achieved. In this investigation, certain operators are to be studied using decomposition techniques sensitive toboth time and frequency considerations. Such operators are often defined by altering the frequency of input signals via multiplication by a fixed and often nonsmooth function. In practice, the abrupt interruption of radio communication or television transmission by a meteorological phenomenon are examples of such operators.The protection against the loss of information can bemathematically modeled as the preservation of integrability and this is proposed to be investigated here.