CAREER: Weighted Fourier Extension Estimates and Interactions with PDEs and Geometric Measure Theory

Overview The PI proposes to study weighted Fourier extension estimates (WFEE), and its variants and applications in PDEs and geometric measure theory. The set of proposed work is closely connected with some important open problems in harmonic analysis, such as the classical Fourier restriction problem raised by Stein. One major special case of WFEE is the L2 to Lp boundedness of the Schrodinger maximal function. With collaborators, the PI has proved sharp results for all L2 to Lp boundedness in dimension 2+1, and for L2 to L2 boundedness in higher dimensions. As an application, such results solved the almost everywhere convergence problem of Schrodinger solutions, which was raised by Carleson about four decades ago. The L2 to Lp boundedness in higher dimensions remains wide open and it is worth further investigation. Another important special case of WFEE applies to Falconer’s distance set problem, which is a famous difficult problem in geometric measure theory and a continuous version of the celebrated Erdos distinct distance conjecture. Using tools from geometric measure theory, the PI and her collaborators observed that one only needs to consider a special case of WFEE for a family of good functions. A promising direction to improve the state- of-the-art results for this problem is to refine the previous work under this additional condition that the functions are good. Another possible method is to carefully remove more bad parts. Intellectual Merit The proposed work will not only apply but also improve various tools and techniques from harmonic analysis, geometric measure theory, incidence geometry, and other related areas. In particular, the work closely relates to several big open conjectures such as Kakeya conjecture, restriction conjecture, Bochner-Riesz conjecture, and Sogge’s local smoothing conjecture. Besides its own independent interest, the study of WFEE in general case has many concrete applications in PDEs and geometric measure theory. Examples are spherical average Fourier decay rates of fractal measures, bounding the size of divergence set of wave equations, etc. Finally, the project will provide researchers in other STEM fields (physics, chemistry, biology, etc) a deeper understanding of the theoretical aspect of Schrodinger equations and quantum theory. Broader Impacts This proposal includes many activities to make mathematical community more inclusive, improve graduate student education, and better serve the Chicago area. The proposal will enrich and support Northwestern Emerging Scholars Program, which aims to increase students from underrepresented minorities in mathematics program at Northwestern through problem sessions, social events, and mentoring. Being a mentor in Math Alliance, the PI will attend its conference to reach out to more underrepresented minorities. The PI will improve harmonic analysis education at her institute by redesigning the curriculum to make the best use of the time in the quarter-system. The PI will support and help organize the Chicago Symposium Series, a forum that engages people from institutions of various types in Chicago area to discuss teaching and research in STEM.