Fractal Fourier Extension Estimates

Project: Research project

Project Details

Description

Overview
The project is devoted to the study of fractal Lp Fourier extension estimates. This type of problems is one of the most important problems in Euclidean harmonic analysis and has interactions with PDE and geometric measure theory. It also has deep connections with the classical Fourier restriction problem raised by Stein, which asks if one can meaningfully restrict Fourier transform f of an arbitary Lp function f to certain subsets with curvature in Rn. An important recent development on restriction/extension problem is the method of polynomial partitioning introduced by Guth. Another breakthrough in close relation is the proof of l2 decoupling conjecture by Bourgain and Demeter. By combining these two methods, the PI (joint with Guth and Li) proved a sharp Schrodinger maximal estimate, which is a special case of fractal extension estimates and has a similar expression as the Fourier restriction conjecture in dimension 3. As an application, this solved the almost everywhere convergence problem of Schrodinger solutions in dimension 2, which was raised by Carleson about 40 years ago. The main novelty in this work is the derivation of linear and bilinear refined Strichartz estimates using decoupling and induction on scales. In another recent work joint with Zhang, the PI obtained fractal L2 estimates, which resolved Carleson’s problem in higher dimensions and provided the state-of-the-art results on Falconer’s distance set problem, spherical average Fourier decay rates of fractal measures, bounding the size of divergence set of Schrodinger solutions, etc. The goal of the proposed study is to make progress towards fully understanding fractal Lp estimates by exploiting ideas from the PI’s previous work as well as developing new tools in a more general setting. There will be applications in multiple problems in analysis.
Intellectual Merit
The fractal Fourier extension estimates have played a key role in the study of numerous analysis problems, and they have close connections with other powerful tools in harmonic analysis such as l2 decoupling and polynomial partitioning. Therefore, a deeper understanding of fractal Fourier extension estimates will yield further progress in many important problems and provide new insights and an integral point of view to several recently developed techniques in harmonic analysis.
Broader Impacts
The project will not only advance modern harmonic analysis and related math branches such as partial differential equations and mathematical physics, but also provide scientists in other STEM fields (physics, chemistry, biology, etc) a deeper understanding of Schodinger equations and quantum theory in general. The project will also provide collaboration opportunities among junior researchers.
StatusActive
Effective start/end date9/1/206/30/22

Funding

  • National Science Foundation (DMS-2107729)

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