Research in Harmonic Analysis

My research lies in harmonic analysis and its interactions with PDE and geometric measure theory. One of my main interests is the Fourier restriction problem raised by Stein, which asks if we can meaningfully restrict the Fourier transform f of an Lp function f to certain subsets such as spheres. It is one of the most important problems in modern Fourier analysis, sitting in the interface of many fields in mathematics, such as PDE, combinatorics, number theory, etc. A very recent development on restriction problem is Guth’s polynomial partitioning method. Another breakthrough in close relation is the proof of l2 decoupling theorem by Bourgain and Demeter. By combining these new tools, joint with Larry Guth and Xiaochun Li, we proved a sharp Schrodinger maximal estimate, which has similar expression as the Fourier restriction conjecture in dimension 3∗. As an application, we solved the 2-dimensional almost everywhere convergence problem of Schrodinger solutions, a problem raised by Carleson about forty years ago. The main novelty in our work is the derivation of linear and bilinear refined Strichartz estimates using decoupling and induction on scales. With Ruixiang Zhang, we also established the fractal L2 Fourier extension estimates†. This implies sharp L2 Schrodinger maximal estimates in all dimensions and hence resolved Carleson’s problem in all dimensions. The proof uses a broad-narrow analysis. In the broad case, we have transversality and can apply either Bennett–Carbery–Tao’s multilinear restriction estimates or multilinear refined Strichartz estimates of Guth, Li, Zhang and myself. In the narrow case, we invoke l2 decoupling in a lower dimension and use a delicate induction on scales argument. Another focus of my research is the Falconer distance set problem, which asks for the minimal Hausdorff dimension required on a compact subset in Rn to ensure that its distance set has positive Lebesgue measure. This open problem has attracted a great amount of attention over the past decades, and the weighted Fourier extension estimates is a very promising approach. The previously best results were built on Wolff and Tao’s work on bilinear restriction estimates. Those records were updated recently by my collaborators and myself. In odd dimensions, the current best result follows from the fractal L2 estimates†. In even dimensions, our method combines various ingredients: refined decoupling inequalities, orthogonal projections of Frostman measures, and Orponen’s radial projection theorem. Currently I have several ongoing projects. One is the multi-parameter version of Falconer’s distance set problem. Another one is to further improve the Falconer’s distance set problem in odd dimensions, trying to combine ideas from fractal L2 estimates and Orponen’s radial projection theorem. I’m also working on the general weighted Lp Fourier extension estimates, which can lead to several applications such as size of divergence set of Schrodinger solutions, Fourier decay rates of fractal measures, etc.