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Fourier Analysis For/And Partial Differential Equations

Periodic Reporting for period 4 - FAnFArE (Fourier Analysis For/And Partial Differential Equations)

Reporting period: 2019-12-01 to 2020-11-30

The project FAnFArE aims to contribute at the interface between the Fourier analysis ans its application to the study of PDEs and/or Geometry.

Two aspects of the modern Fourier Analysis are mainly investigated:

1) The Euclidean Fourier Analysis, where a deep analysis can be performed using specificities as the notion of ``frequencies'' (involving the Fourier transform) or the geometry of the Euclidean balls. By taking advantage of them, this proposal aims to pursue the study and to bring novelties in three fashionable topics : the study of bilinear/multilinear Fourier multipliers, the study of dispersive PDE, and the study of nonlinear transport equations in BMO-type spaces (as Euler and Navier-Stokes equations).

2) A Functional Fourier Analysis, which can be performed in a more general situation using the notion of ``oscillation'' adapted to a heat semigroup (or semigroup of operators). This second challenge is (at the same time) independent of the first one and also very close. It is very close, due to the same point of view of Fourier Analysis involving a space decomposition and simultaneously some frequency decomposition. It is technically different from the first challenge, since one is supposed to develop an analysis in a more general framework given by a semigroup of operators. By this way, we aim to transfer some results known in the Euclidean situation to some other frameworks.

In addition, the project aims also to contribute in building a network of young and promizing mathematicians around these topics, by organizing workshops and hiring postdocs.
Reaching the end of the project, I succeed to constitute a team by hiring postdocs and we have made progress on the two main aspects of the project. As explained, the project relies on recruiting young analysts as postdoc and so the progress and the organization of the whole project also depends of the candidates, in the job market for a postdoc position.

The main activity in pure mathematics relies on academic research which gives publications and the organization/participation of events, which is what has been done during and through project FAnFArE.

In terms of mathematical results, we have tried to follow the tasks explained and planned in the DoA. Answers to the initial questions have been obtained. Moreover, allowing to bring a better understanding, some investigation in new and very promising directions, in connexion with the topic of the project have also been conducted, At the end, the project has given rise to different kind of collaborations, invitations, visits of experts, achieved through research papers published in international journals. The results cover all the different six tasks. So mainly, project FAnFArE has succeed to contribute in the study of multilinear Fourier multipliers in the Euclidean space, by describing boundedness of some certain class of them. It has been related to the notion of sparse domination, vector-valued estimates, ... The team has also started the study of several objects in the multi-parameter setting and in a more geometrical setting involving some curvature. A deep understanding of analytical objects allow us to have sharp estimates and to prove sharp inequalities whose one of the consequence is the study of several kind of PDE (dispersive, fluids mechanic, ...). The time-frequency analysis associated to a semigroup of operators, which allows to tackle more abstract situations (as on a Riemannian manifold), has also been developed. With such a setting, we have focused on the study of several kinds of paraproducts according to the problem under consideration. That allows us to study the algebra property of Sobolev spaces (for a positive regularity) and to study singular (stochastic) PDE (for a negative regularity) for example.

So as expected and as planned initially, together the team FAnFArE has made important contributions in the time-frequency analysis both in the case of the Euclidean case (with a deep analysis for specific singular operators) and in the more abstract setting given by a semigroup of operators.

In parrallel of these mathematical achievements, several workshops / conferences have been organized, as well as various participations to international conferences in order to promote the project and the obtained results. The list of these events is updated on the website of the project :
Each year of the project, I have organized (or co-organized) with other members of the team, at least one main event focusing on inviting young promising analysts and more experienced experts of our field. By these events, the project is promoted and it allows to make connections and develop collaborations together, taking advantage of the skills of every members of the team.
During the project, I aim to develop a network of young researchers in Fourier Analysis. I keep updated a website on the ERC project, where we list the publications, the members, as well as the organized events and the visitors.

As usual in mathematic research, the main activity is to participate to conference and organize visits or invitations of experts. Through these activities, the team is collaborating with the worldwide mathematical community.

The project FAnFArE has an important impact in the group of young mathematicians in Fourier Analysis. I had the opportunity to hire four postdoc students, all of them having obtained another position in Europe, at the end of their participation in the project. So each of them and myself, we will continue to work together, as well as to disseminate and build a larger group of collaboration.
Harmonic Analysis