Problems in harmonic analysis

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ABSTRACTThe principal investigator is planning to work on a variety of problems in harmonic analysis arising in the study of multilinear and rough linear singular integral operators. These problems are related, but not limited, to the study of the Carleson operator and the bilinear Hilbert transform. A significant breakthrough was made by the inspiring work of Lacey and Thiele on the bilinear Hilbert transform. This work instigated renewed interest and activity in the area of multilinear singular integral operators and also in the underlying tool of study of these problems, time-frequency analysis. This method is relevant to some of the principal investigator's work as he is planning to work on some problems in this proposal using time-frequencyanalysis. These problems include uniform bounds for the bilinearHilbert transform in the full range of exponents possible, the study of the disc and the corresponding maximal bilinear multiplier, the study of the Hilbert transform along Lipschitz vector fields, and the study ofCarleson-type operators arising in the almost everywhere convergence of spherical partial sums of Fourier series in higher dimensions.Recently, M. Lacey and the PI had been able to use the time-frequencyanalysis to obtain an affirmative result for the Hilbert transform along $C^{1+\e}$ vector fields. And this casts a light on answering the Zygmund conjecture, which asks if the integrals of $L^2({\mathbb R}^2)$ functions could be differentiated in a Lipschitz choice of directions.Although the time frequency analysis works well for multiliner singularintegrals associated with a standard Carderon-Zygmund kernel, it does not work for the nonsingular oscillatory integrals. M. Christ, T. Tao,C. Thiele, and the PI established $L^p$ estimates for the multilinear oscillatory integrals. The proof relies on Gowers ``quadratic uniformity'' technique, which is a crucial concept in Gowers' proof of Szemeredi's theorem on arithmetic progression of length 4. Some basic questions concerning nonsingular multilinear operators with oscillatory factors are still under the investigation.The differentiability of the integral of certain functions is a veryimportant and interesting subject in analysis. The construction of Besicovitch set indicates that integrals of L^2 functions can not be differentiated. The recent results obtained by M. Lacey and the PI have a significant impact on understanding the differentiablity of $L^2({\mathbb R}^2)$ functions in a Lipschitz choice of directions, which was posed by A. Zygmund about seventy years ago. The study of Hilbert transform along vector fields is also relevant to the Kakeya problem. The study of multilinear operators is related to some problems in PDE such as the Schrodinger equation