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 $R^3$
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 $(n,n^2)$. 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 $R^3$. Our work further sharpens the refined $L^4$ square function estimate for the truncated cone $C^2=\{ (x,y,z) \in R^3: x^2+y^2 = z^2, 1/2 \leq |z| \leq 2 \}$ from the local smoothing paper of L. Guth, H. Wang, and R. Zhang. A corollary is sharp $(\ell^p,L^p)$ small cap decoupling estimates for the cone $C^2$, for the sharp range of exponents p. The base case of the “induction-on-scales” argument is the corresponding sharpened, refined $L^4$ 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.