Harmonic Analysis and PDEs Meeting
18th of March, 2021
Zoom meeting ID: 825 9963 0625
The LMS Harmonic Analysis and PDE Network has been running since 2005 and is currently coordinated by María Carmen Reguera.
The participating universities are Bath, Birmingham, Edinburgh, Kent and Warwick in the UK, and the Universidad Autónoma de Madrid and the Instituto de Ciencias Matemáticas in Spain.
The 18th March meeting is meant as an opportunity for three early career researchers to present their original work. It is open to all interested participants, seniors and juniors alike, and it will be hosted online on the Zoom platform, at the ID 825 9963 0625. A passcode is required to access the meeting; if you haven't received it, you can email Giuseppe Negro
or Tomás Sanz-Perela
One of the purposes of this meeting is to facilitate communication and socialisation between participants. Thus, after the talk there will be an informal time on breakout Zoom rooms.
Inequalities that carry some kind of transversality condition are ubiquitous in multilinear harmonic analysis, among which the Brascamp-Lieb inequalities are fundamental examples. In this talk, we shall discuss some of the history of the topic, focusing in particular on nonlinear Brascamp-Lieb inequalities, and how it relates to other areas in harmonic analysis.
One of the main properties of functions of Bounded Mean Oscillation is the self-improvement of their integrability, a fact that is encoded within the John-Nirenberg theorem. In this talk, we will present an estimate for the Hardy-Littlewood maximal function that generalizes the JN theorem. This estimate can be applied to obtaining sharp weighted estimates between the Hardy-Littlewood and the sharp maximal functions. We will also discuss minimal conditions for defining BMO. These conditions describe weak local oscillations that are measured with concave functions via Luxemburg-type expressions.
This talk is based on joint work with Carlos Pérez (UPV/EHU and BCAM) and Ezequiel Rela (Universidad de Buenos Aires).
In this talk, we consider the periodic generalized Korteweg-de Vries equations (gKdV). In particular, we study gKdV with the Gibbs measure initial data. The main difficulty lies in constructing local-in-time dynamics in the support of the measure. Since gKdV is analytically ill-posed in the -based Sobolev support, we instead prove deterministic local well-posedness in some Fourier-Lebesgue spaces containing the support of the Gibbs measure. New key ingredients are bilinear and trilinear Strichartz estimates adapted to the Fourier-Lebesgue setting. Once we construct local-in-time dynamics, we apply Bourgain's invariant measure argument to prove almost sure global well-posedness of the defocusing gKdV and invariance of the Gibbs measure.
Our result completes the program initiated by Bourgain (1994) on the invariance of the Gibbs measures for periodic gKdV equations.
This talk is based on joint work with Nobu Kishimoto (RIMS, University of Kyoto).
Schedule (in GMT - add 1 hour for the time in Madrid):
14:00 Jennifer Duncan
15:15 Javier Canto
16:30 Andreia Chapouto
17:30 Social time
If you have any questions, please feel free to contact Tomás Sanz-Perela
or Giuseppe Negro