CE40 - Mathématiques, informatique théorique, automatique et traitement du signal

Real Analysis and Geometry – RAGE

In order to attack a number of problems at the interface of Analysis and Geometry, the method is to bring together in this project experts and young researchers working on hamonic analysis, functinal analysis, PDE and geometry. For example, in order to understand the problem of the Riesz transform we try to better understand the heat kernel on functions and on differential forms as well as Hardy spaces on differential forms on manifolds. We also aim to develop new tools for maximal regularity in order to attack boundary value problems for parabolic operators.

We have a large number of interesting results which are described in the attached report. We quote here few of them: description of Hardy spaces on manifolds with Ricci curvature having quadratic decay, spectral multipliers for operators with heat kernel having a slow decay, new Littlewood-Paley-Stein functionals, interpolation inequalities and Bakry-Emery curvature type in sub-Riemannian geometry, spectral estimates for Schrödinger operators on manifolds satisfying relative Faber-Krahn inequalties...

We continue our research on the problems described in the project. As explained there some of the problems are of exploratory character and need more time. Because of the health crisis we were not able to organize meetings during 2020 but we had online ones. As a consequence, collaboration between members of differents universities need to be improved. It is one of our aims for the next part of the project to encourage such collaborations.
From the scientific point of view, the project is running very well.

Here is a list of publications. There is a large number of preprints (see the attached report):
-Mietton and Rizzi, Branching geodesics in sub-Riemannian geometry, GAFA, 2020, hal-02493682
- Rizzi and Rossi, Heat content asymptotics for sub-Riemannian manifolds, J. Math. Pures Appl. 2021. hal-02563090
- Barilari and Rizzi, Bakry-Émery curvature and model spaces in sub-Riemannian geometry, Math. Ann., hal-02163180
-Rossi, Integrability of the sub-Riemannian mean curvature at degenerate characteristic points in the Heisenberg group, 2020, Adv. Calc. Var. hal-02960528
- Moonens, Russ, Solvability in weighted Lebesgue spaces of the divergence equation with measure data, Studia Math. Hal-02508132.
-Devyver, Russ, Hardy spaces on Riemannian manifolds with quadratic curvature decay, Anal. PDE, Hal-02320652.
- Carron Geometric inequalities for manifolds with Ricci curvature in the Kato class, Ann. Inst. Fourier, 2020. Hal-01441891.
- Carron, Euclidean volume growth for complete Riemannian manifolds, Milan J. Math. 2020. Hal 02502010.
-Carron and C. Rose, Geometric and spectral estimates based on spectral Ricci curvature assumptions, J. reine und angew. Math. 2021.
- Egert, On p-elliptic divergence form operators and holomorphic semigroups. J. Evol. Eqs, 2020, Hal-01961907
- Bechtel, M. Egert, R. Haller-Dintelman, The Kato square root problem on locally uniform domains, Adv. Math. 2020.
- Cometx, Littlewood-Paley-Stein Functions for Hodge-de Rham and Schrödinger Operators. J. Geom. Anal, 2021. Hal-02416797.
- Chen, Ouhabaz, Sikora and Yan, Spectral multipliers without semigroup framework and application to random walks. J. Math. P. Appl. 2020. Hal-02072126.
- Bonnefont, Golenia, Keller, Liu and Munch, Magnetic sparseness and Schrödinger operators on graphs. Ann. Henri Poincaré (2020)
- Bonnefont and Juillet, Couplings in Lp distance of two Brownian motions and their Levy area. Ann. Inst. Henri Poincaré Probab. Stat. (2020). Hal-01671676

Submission summary

Real analysis in non Euclidean contexts such as Lie groups, (sub)-Riemannian manifolds, graphs, fractals or more general metric spaces has evolved quite spectacularly in the recent years. At the same time, a huge amount of work was done on analysis of differential operators with non-smooth coefficients. In these subjects, the properties of the heat semigroup on functions or on differential forms played an essential role. On the other hand, it is crucial to understand the geometrical aspects that are involved when considering problems outside the Euclidean context. There are several challenging problems in which analysis and geometry are intimately linked, and an extremely good knowledge of both aspects is necessary to attack them. This is what we wish to do in the framework of this ANR project. We intend to gather well-known experts and young researchers in analysis and geometry in order to tackle new problems at the interface of these two directions. We develop further connections between real analysis and geometry. We shall investigate the heat kernel on Riemannian manifolds together with bounds on its spacial gradient, the heat kernel and heat semigroup on 1-différential forms (the semigroup of the Hodge-de Rham operator), L^p-boundedness of the Riesz transform, Sobolev and Besov algebras in the setting of sub-Gaussian bounds, the heat kernel and functional inequalities in the setting of sub-Riemannian geometry (such as the Heisenberg group), spectral estimates of Schrödinger operators, bi-parameter analysis, spectral multipliers, parabolic equations/systems, boundary value problems, maximal regularity for non autonomous equations... Most of these problems are at the interface of real analysis and geometry.

Project coordination

El Maati Ouhabaz (Institut de mathématiques de Bordeaux)

The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.


IF Institut Fourier
IMB Institut de mathématiques de Bordeaux

Help of the ANR 199,704 euros
Beginning and duration of the scientific project: December 2018 - 48 Months

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