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Analysis Seminar

Thursday, January 31, 2019 - 3:00pm

Xiumin Du

University of Maryland, College Park


University of Pennsylvania


We consider Carleson’s pointwise convergence problem of Schrodinger solutions. It is shown that the solution to the free Schrodinger equation converges to its initial data almost everywhere, provided that the initial data is in the Sobolev space $H^s(\mathbb{R}^n)$ with $s > n/2(n+1)$ (joint with Larry Guth and Xiaochun Li in the case $n=2$, and joint with Ruixiang Zhang in the case $n>2$). This is sharp up to the endpoint, due to a counterexample by Bourgain. This pointwise convergence problem can be approached by estimates of Schrodinger maximal functions. The key ingredients are refined Strichartz type inequalities derived from Bourgain–Demeter decoupling theorem and induction on scales. This technique can also be used to obtain the state-of-art results on many other problems such as the size estimate for divergence set of Schrodinger solutions, the Falconer distance set problem and the spherical average Fourier decay rates of fractal measures.