Problems in Harmonic Analysis

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Proposal Number: DMS-0140376PI: Xiaochun Li ABSTRACTResearch will be conducted on a variety of problems in harmonic analysis arising in the study of multilineasingular integral operators. These problems are related,but not limited, to the study of the Carleson operator and the bilinear Hilbert transform. Recently a significantbreakthrough on the bilinear Hilbert transform was made by the inspiring work of Lacey and Thiele. It turns out that the time-frequency analysis is a powerful tool to solve problems related to the study of multilinear operators. Moreover, although this is still under investigation, the analysis of these problems gives us hopes to solve other important and difficult problems in the field, such as the Hilbert transform along vector fields, the Kakeya problem, and Carleson's maximal operator of the partial sums of Fourier series in two dimensions. It will also be very interesting to see how to use this delicate analysis to solve some problems in other fields such as partial differential equations, number theory, etc. Actually, multilinear operators have been used in the study of partial differential equations, since they naturally appear in series expansions of solutions of many equations. Harmonic analysis is not only an area of theoretical mathematics, but also an applicable area lying at theheart of the intersection of fields as diverse as optics, signal processing, meteorology, and music. The main themin Harmonic analysis is about disassembling and assemblingcomplicated objects into simpler well-understood pieces, by analogy to decomposition of intricate musical pieces into arrangements of a few basic notes. In signal processing,harmonic analysis is used to detect irregularities of signals and images. The Fouriertransform is a very useful tool to locate these irregularities. The appearance of a nonsmooth symbol inthe study of multiplier problems is analogous to physical phenomena where the frequencies of signals are altered by an abrupt operation, such as the interruption of radio communication or television transmission by meteorological phenomena. Such a sudden and unexpected operation causes the loss of information. To avoid the loss of information or to retrieve the original data is the main topic of the theoretical research proposed here.