CAREER: Weighted Fourier extension estimates and interactions with PDEs and geometric measure theory

Northwestern University, Evanston IL

Investigators

Abstract

Harmonic analysis is an important branch of mathematics. A key idea behind harmonic analysis is to express a general function or operator as a sum of simpler parts. Harmonic analysis has countless practical applications in signal processing, tomography, quantum mechanics, etc. It is also a powerful tool to study many theoretical aspects of mathematics. Fourier restriction theory is a subfield of harmonic analysis, which asks if one can meaningfully restrict the Fourier transform of a function onto a hypersurface, for example, a sphere. One then studies how small pieces of a function with different frequencies interfere with each other. Fourier restriction theory is a central topic in harmonic analysis and plays a fundamental role in certain problems in number theory, differential equations, and geometric measure theory. Thanks to the development of new ideas and techniques in restriction theory, several new state-of-the-art results in harmonic analysis and related fields have been established recently. However, a large portion of the new results are still not sharp, or are unknown in the general dimensions. This project will further develop these new ideas and techniques in restriction theory, and push forward the current best results for related questions, especially in the high-dimension case. The project’s multifaceted activities will include a new component for the existing Northwestern Emerging Scholars Program, work with the Math Alliance, involvement in the Chicago Symposium series, as well as the initiation of a summer program: Harmonic Analysis Reading and Research in Summer (HARRIS). More specifically, in the project weighted Fourier extension estimates (WFEE), and their variants and applications in partial differential equations and geometric measure theory, will be studied. One important case of WFEE is the boundedness of the Schrödinger maximal function. Such estimates are motivated by the recent proof of the almost everywhere convergence problem of Schrödinger solutions, a question which was raised by Carleson four decades ago. The full range of boundedness of the Schrödinger maximal function has been established when the spatial dimension is 1 or 2, but it remains open in higher dimensions. Another special case of WFEE that will be investigated is connected to a difficult problem in geometric measure theory: Falconer's distance set problem. The project will not only apply but also improve various tools and techniques from harmonic analysis, geometric measure theory, incidence geometry, combinatorics, and other related areas. 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.