Topics in Harmonic Analysis

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Methods from the mathematical discipline of analysis have found wide applications in understanding physical phenomena in the natural sciences and engineering. This project is concerned with topics in harmonic analysis that are designed to provide efficient mathematical tools for these disciplines and that are expected to contribute to the unification of seemingly unrelated areas. Of particular interest is the study of various integral transforms, such as the Fourier and the X-ray transforms, and some of their relatives. The Fourier transform decomposes a signal (mathematically a function) into the frequencies that make it up. It has been found to be relevant for solving many mathematical problems arising in science and engineering and can be regarded as a special case of a larger class of oscillatory integral operators. The X-ray transform is an operator that assigns to a function its integral over lines. It is relevant to problems in medical imaging and can be considered as a special case of a larger class of Radon type transforms and averaging operators. A main goal of the project is to expand the current mathematical toolbox in harmonic analysis to contribute towards a deeper theoretical understanding of these integral transforms and their generalizations.The principal investigator will work on several projects in harmonic analysis. The first project is concerned with the application of decoupling theory to regularity questions for averaging operators and generalized Radon transforms, and to boundedness problems for associated maximal functions. The second project deals with a new multiplier problem for Haar expansions in spaces of Hardy-Sobolev type, in ranges where the Haar basis is not unconditional. The third project investigates spectral multipliers for the Kohn Laplacian on the Heisenberg group and the behavior of the corresponding multiplier transformations on Lebesgue spaces. It is proposed to prove new space-time estimates for solutions of the wave equation associated with the Kohn Laplacian and to use them to bound the multiplier operators. The principal investigator will also work on other projects, related to almost everywhere convergence of Bochner-Riesz means, local improving inequalities for maximal functions with application to sparse domination results, and boundedness of multilinear singular integral operators. The project involves mentoring of graduate students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.