Analyse Réelle et Géométrie – 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.