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Fourier Analysis For/And Partial Differential Equations

Periodic Reporting for period 3 - FAnFArE (Fourier Analysis For/And Partial Differential Equations)

Reporting period: 2018-06-01 to 2019-11-30

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

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. 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 mathematician around these topics, by organizing workshops and hiring postdocs.
"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 (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). Then, we have also recruited Teresa Luque for a postdoc (6 months) and then she got a position at Universidad Complutense de Madrid. We have then recruited Roberto Feola (for 5 months and he has continued for a 2 years postdoc position, here in Nantes) and Marco Vitturi for a postdoc position of two years (he is now postdoc in Cork University - Ireland). Finally, to conclude the project, we have hired Yujia Zhai for a postdoc position of two years (with the financial support of the region Pays de la Loire for the second year, the first year beeing support by the erc project).

The main activity in pure mathematics relies on academic research which gives publications and the organisation/participation of events.

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Let us first summarize the different works, according to the different tasks of the project (the reference of the publication correspond to the one in the list of publications).

Task 1A) : This task was devoted to the study of the bilinear time-frequency analysis, in order to develop and improve the understanding of boundedness of singular bilinear Fourier multipliers. In this context, Benea and Muscalu have pursued the development of the ""helicoidal method"", that they have introduced, in [6,12,13,15,21]. This theory allows to obtain an abstract framework where multilinear Fourier multipliers are shown to satisfy new mixed Lebesgue norms, multiple vector-valued norms, sparse domination, weighted estimates and as application they in [17] obtained some lower estimates for the Littlewood-Paley square function.

In [2], Benea and Bernicot have also initiated the study of a ""bilinear orthogonality principle"": we consider an arbitrary collection of squares in the frequency plane and associated to it, we build some bilinear functionals and study their boundedness in Lebesgue spaces. We obtained a first result for smooth truncations and then this result was generalized by Bernicot and Vitturi in [14] for non-smooth cutoff and then in [16] by considering an arbitrary collection of rectangles.

Task 1B) : In the context of dispersive PDEs, Germain, Gérard and Thomann have continued their collaboration around the dynamics of some nonlinear dispersive estimates for large time [9] and Feola with his collaborators has studied some qualitative behavior of solutions of PDEs on compact manifolds [18,19,20].

Task 1C) : Frédéric Bernicot and Yujia Zhai have started the study of the composition in the biparameter BMO space, as well as the understanding from a transport equation point of view of the propagation in the BMO space. That should allow us to obtain some results for the 3D Euler equation in some BMO type spaces.

Task 2A) : Valentin Samoyeau (who was in PhD at University of Nantes), has pursued our investigation about dispersive estimates through the heat semigroup [3].

Task 2B) : Associated to semigroup, Benea and Bernicot have obtained in [11] an abstract approach to prove sparse domination for singular operators as Riesz transform, Leray projector and obtaining by this way new end-point estimates.

Task 2C) : With Frey and Bailleul, Bernicot has developed a suitable notion of paraproducts through semigroup to understand the singular PDEs, as well as a higher order paracontrolled calculus in order to solve singular / stochastic PDEs in a very abstract setting [5,7].
With Frey, Bernicot has studied Sobolev Algebras, through semigroups and an extra geometrical property related to a carré du champ structure [8].

Moreover, following a"
During the project, we aim to develop a network of young researchers in Fourier Analysis. 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.

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

The project FAnFArE has an important impact in the group of young mathematicians in Fourier Analysis.
Harmonic Analysis