This talk is a joint event with the AIM Research Community “Fourier restriction conjecture and related problems“.

Prof. Larry Guth has also kindly agreed to host an office hour taking questions and discussing more about the content of the talk with interested viewers. The office hour will be held on Wednesday, March 16, from 3pm-4pm EDT (

Abstract: The goal of this talk is to explore some of the difficulties involved in an open problem in restriction theory. The problem is the Mizohata-Takeuchi conjecture for the parabola, which is a special case of a conjecture Stein raised. This is a conjecture about weighted estimates for the extension operator for the parabola. While we focus on this problem for concreteness, I think that there are similar difficulties involved in a number of open problems in restriction theory. A lot of work in restriction theory leans heavily on wave packet decompositions, and on a short list of basic properties of wave packet decompositions. For instance, the proof of decoupling for the parabola (or paraboloid or moment curve) only uses this list of basic properties. We will describe an enemy scenario for the M-T conjecture. This scenario would respect the basic properties of wave packet decompositions used in decoupling, but would violate the M-T conjecture. I don’t know whether this scenario actually happens. In order to rule out this scenario, we would have to find a new tool.