Problems in Harmonic Analysis

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The 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. Some problems that the PI is planning to work are related to time-frequency analysis, which was used in the celebrated Lacey-Thiele's theorem on the bi-linear Hilbert transform. These problems include uniform bounds for the bi-linear Hilbert transform, the disc and the corresponding maximal bi-linear multiplier, the Hilbert transform along Lipschitz vector fields, and multi-linear Carleson-type operators arising in the almost everywhere convergence of spherical partial sums of Fourier series in higher dimensions. Some problems are related to Szemer\'edi's theorem on arithmetic progression and the (multilinear) oscillatory integrals, in which the time frequency analysis is unvaluable. These type of problems contain the bi-linear Hilbert transform along curves, and the multi-linear oscillatory integrals along curves, the muliti-linear oscillatory integrals with non-degenerate phases which was first studied by Christ, Tao, Thiele and the PI. Some problems are associated to the Kakeya problem such as the Lipschitz maximal functions, initiated by Lacey and the PI, and the Zygmund conjecture. Recently, C. Muscalu and the PI generalized the Carleson-Hunt theorem to the multi-linear case. And M. Lacey and the PI had been able to use the time-frequency analysis to obtain some conditional results for the Hilbert transform along Lipschitz vector fields. A fundamental subject in analysis is the differentiability of the integral of certain functions. Recently M. Lacey and the PI proved some estimates for a Lipschitz Kakeya maximal function by a completely new method based on some crucial geometric observations. It turns out that a complete understanding of the Lipschitz Kakeya maximal function is a key to answer the question on the differentiablity of certain functions in a Lipschitz choice of directions, which was posed by A. Zygmund about seventy years ago. The Hilbert transform along curves had been understood well by work of many people in the last several decades. However, the bi-linear Hilbert transform along curves is a new field. Some partial results such as uniform estimates for some para-products arising in the study of this type of problem were obtained by the PI. As the linear case, the relation of multi-linear singular integrals along curves and multi-linear oscillatory integrals is an interesting and important topic in analysis. This relation is only partially understood. Based on it, D. Fan and the PI obtained an affirmative result for the bi-linear oscillatory integral along parabolas incorporating some oscillatory factors, which is a starting point for understanding the bi-linear Hilbert transform along curves. The main theme in harmonic analysis is disassembling and assembling complicated objects into simpler well-understood pieces, called frequencies, by analogy to decomposing musical pieces into arrangements of a few basic tones. In signal processing, harmonic analysis is used in the detection of irregularities of signals and images, the protection against the loss of information due to a sudden and unexpected interruption, and the retrieval of the original data. These applications provide the main practical motivation for some theoretical research described in this proposal. The research experience and results gained by investigating these difficult problems such as the Stein's and Zygmund's conjectures help the PI better understand a wide range of topics. It also provides the PI with some new visions on mathematics that greatly benefits the PI's teaching. Hopefully it will also benefit fellow mathematical educators, and more importantly dedicated mathematical students, through sharing the new knowledge and visions.