Dominique Maldague (MIT)

The APDE seminar on Monday, 11/22, will be given by Dominique Maldague (MIT) both in-person (740 Evans) and online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (sjoh@math.berkeley.edu).

Title: Small cap decoupling for the cone in R3

Abstract: Decoupling involves taking a function with complicated Fourier support and measuring its size in terms of projections onto easier-to-understand pieces of the Fourier support. A simple example is periodic solutions to the Schrodinger equation in one spatial and one time dimension, which have Fourier series expansions with frequency points on the parabola (n,n2). The exponential sum (Fourier series) itself is difficult to understand, but each summand is very simple. I will explain the basic statement of decoupling and the tools that go into the most current high/low frequency approach to its proof, focusing on upcoming work in collaboration with Larry Guth concerning the cone in R3. Our work further sharpens the refined L4 square function estimate for the truncated cone C2={(x,y,z)R3:x2+y2=z2,1/2|z|2} from the local smoothing paper of L. Guth, H. Wang, and R. Zhang. A corollary is sharp (p,Lp) small cap decoupling estimates for the cone C2, for the sharp range of exponents p. The base case of the “induction-on-scales” argument is the corresponding sharpened, refined L4 square function inequality for the parabola, which leads to a new proof of canonical (Bourgain-Demeter) and small cap (Demeter-Guth-Wang) decoupling for the parabola.