# Mathematical Sciences Research Institute

Home » Graduate Student Working Group: The Fourier Extension problem through a time-frequency perspective

# Seminar

Graduate Student Working Group: The Fourier Extension problem through a time-frequency perspective April 07, 2021 (11:10 AM PDT - 12:10 PM PDT)
Parent Program: Mathematical problems in fluid dynamics MSRI: Online/Virtual
Speaker(s) Itamar Oliveira (Cornell University)
Description

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Video

#### The Fournier Extension Problem Through a Time-Frequency Perspective.mp4

Abstract/Media

To participate in this seminar, please register here: https://www.msri.org/seminars/25657

Itamar Oliveira (Cornell University)

Title: The Fourier Extension problem through a time-frequency perspective

Abstract: An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $L^{2+\frac{2}{d}}([0,1]^{d})$ to $L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1})$ for every $\varepsilon>0$. It has been fully solved only for $d=1$ and there are many partial results in higher dimensions regarding the range of $(p,q)$ for which $L^{p}([0,1]^{d})$ is mapped to $L^{q}(\mathbb{R}^{d+1})$. In this talk, we will take an alternative route to this problem: one can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for tensor functions, meaning for all $g$  of the form $g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d})$. Time permitting, we will also address multilinear versions of the statement above and get similar results, in which we will need only one of the many functions involved in each problem to be of such kind to obtain the desired conjectured bounds. This is joint work with Camil Muscalu.

Lizhe Wan (University of Wisconsin, Madison)

Title: The $L^2$ well-posedness of Benjamin-Ono equation

Abstract: The Benjamin-Ono equation, a model for the propagation of one-dimensional internal waves, is a nonlinear dispersive equation with many interesting properties. In this talk, I will first present a proof of the $L^2$ well-posedness of the Benjamin-Ono equation. The main idea is based on the partial normal form transformation together with the renormalization. I will then briefly talk about the long time decay property of the Benjamin-Ono equation. This is the work of Mihaela Ifrim and Daniel Tataru.