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FAnFArE Report Summary

Project ID: 637510
Funded under: H2020-EU.1.1.

Periodic Reporting for period 1 - FAnFArE (Fourier Analysis For/And Partial Differential Equations)

Reporting period: 2015-06-01 to 2016-11-30

Summary of the context and overall objectives of the project

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

We are mainly interested in two aspects of the modern Fourier Analysis:

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 development of the "space-time resonances" method in a systematic way and for some specific PDEs, 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. However they are quite independent because the main goal is to extend/develop an analysis in the more general framework given by a semigroup of operators (so without using the previous Euclidean specificities). By this way, we aim to transfer some results known in the Euclidean situation to some Riemannian manifolds, Fractals sets, bounded open set setting, ... Still having in mind some applications to the study of PDEs, such questions make also a connexion with the geometry of the ambient spaces (by its Riesz transform, Poincaré inequality, ...). I propose here to attack different problems as dispersive estimates, Lp-version of De Giorgi inequalities and the study of paraproducts, all of them with a heat semigroup point of view. Such paraproducts can also be used to extend the paracontrolled calculus in order to study singular / stochastic PDEs in a general setting.

Also one of the main objective of the project is to contribute in building a network of young mathematician around these topics, by organizing workshops, as well as financially supporting postdoc positions. In that point of view, this project can be seen as an investissment for the future.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Up to today, we have made progress on several aspects in 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 on the candidates, in the job market for a postdoc position.

We have first employed Cristina Benea as a postdoc for 1 year (and she has then obtained a permanent position at University in Nantes, where the project is located and so she is pursuing her collaboration) from 09/2015 to 08/2016. With her, we have worked on the first task (Task I.A) of the project (about bilinear time-frequency analysis) and we have obtained first results which have given a publication: we have developed new arguments to understand the bilinear square function associated with a covering of the frequency plane, where we needed to deal with an arbitrary geometry in the frequency. With Camil Muscalu, she has also continued the study of bilinear vector-valued estimates, which has given two publications, where an improvment of the bilinear time-frequency analysis has been developed.

In November, I have also visited Pierre Germain in New-York, in order to continue our collaboration fitting in Task I.B of the project. In particular, we have initiated investigation in the study of nonlinear dispersive PDEs and combining with the new paracontrolled approach to understand how we can combine the dispersive properties and some random phenomenas.

With Valentin Samoyeau (who was in PhD at University of Nantes), we have pursued our investigation about dispersive estimates through the heat semigroup (Task II.A). This is a fully original topic and very difficult, which relies to develop new arguments to understand the dispersive properties (which is well-known to be crucial in the study of PDEs) in an abstract setting given by a semigroup. The main idea is to obtain a new approach of dispersive properties, through analytical properties of the heat semigroup. This work was also submitted for a publication.

We have also recruited Teresa Luque as a postdoc from 05/2016 to 09/2016 (she has then obtained a permanent position in Madrid) and jointly with Cristina Benea, we have worked on a novel aspect of the modern Fourier analysis: the sparse technology. This theory is very recent and was not clearly expected in the current project. However, we have thought that this very powerful technology could be very interesting to us, to make progress and make the project successful. So we decided to learn and we have obtained first results for Bochner-Riesz means (which are a particular of Fourier multipliers with a specific singularity), which have been submitted for a publication. With Dorothee Frey and Stefanie Petermichl, we have also extended this sparse technology to Riesz transform, adapted with a heat semigroup. This is closely related to the Task II.B of the project, in order to understand sharp weighted estimates for such operators and was published.

With Dorothee Frey and Ismael Bailleul, we have also worked on the Fourier analysis associated with a semigroup, in order to develop a suitable notion of paraproducts to understand the singular PDEs (Task II.C). This work has given two publications and allows us to give an important contribution in the very recent theory of paracontrolled calculus, in order to give a sense and solve singular / stochastic PDEs in a very abstract setting given by a semigroup.

With Dorothee Frey, we have also continued our collaboration about the study of Sobolev Algebras, through the study of heat semigroup, in a new publication we have improved previous results assuming an extra geometrical property related to a carré du champ structure (task II.C).

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

During the project, we aim to develop a network of young researchers in Fourier Analysis, through several aspects. We try to keep updated a website on the ERC project, where we list the publications, the members, as well as the events that we organize and the visitors.

We organize different workshop / conferences in order to promote our results and learn other mathematics from the worldwide community. We have organized in June 2016 a summer school (one week) in Harmonic Analysis, jointly with a french project ANR HAB. This was very successful. We aim also to organize in March 2017 a workshop for young analysts, in order to make connections and develop collaborations together, taking advantage of the skills of every one. Other conferences are expected to be organized (one in CIRM / Marseille in October 2017 and another workshop in time-frequency analysis / sparse technology in a CNRS center in March 2018).
By these events, the project is promoted and it allows us to detail our results, as well as to stay aware of what is done in the world.

We also offer several postdoc positions. We had first Cristina Benea (1 year), Teresa Luque (5 months) who have both obtained permanent positions after having worked through the erc project. A new collaborator is expected to come for a postdoc of 2 years, Marco Vitturi from January 2017. So that illustrates how we are able to attract very talented young mathematicians and that the research in Fourier Analysis is very fashionable.

Also, one of the main activity in the mathematical research is to participate to conference / workshop and financially cover trips to visit or invite experts. Through these activities, we make the team collaborating with the worldwide mathematical community, and participating to international conferences (in Europe and in USA).

By all these aspects, the project FAnFArE has an important impact in the group of young mathematicians in Fourier Analysis and is a tool for investing in this topic.
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