Séminaire d'analyse (archives)

La transformée de Riesz $\nabla \Delta^{-1/2}$ sur $\mathbb R^n$ peut être définie comme un opérateur integral, dont le noyau est de Calderon-Zygmund, ce qui implique que $\nabla \Delta^{-1/2}$ est continu sur $L^p(\mathbb R^n)$ pour tout $1<p<+\infty$. Dans les années 80, Strichartz se demanda si cette propriété sur la transformée de Riesz est transmise aux variétés riemanniennes, plus exactement, quelles sont les conditions géométriques nécéssaires ou suffisantes sur une variété riemannienne qui impliquent la continuité $L^p$ de la transformée de Riesz.

J'exposerai (une partie) de la littérature sur le sujet, puis je présenterai notre résultat (en collaboration avec Li Chen, Thierry Coulhon, et Emmanuel Russ) sur des variétés riemanniennes de type fractal (i.e. avec un noyau de la chaleur qui vérifie des estimations sous-gaussiennes). Je parlerai aussi du cas des graphes, qui peuvent être vus comme la version discrète des variétés riemaniennes, ce qui me permettra de donner des exemples d'application de nos résultats.

Si le temps le permet, je ferai un lien entre l'analyticité discrète (sur $L^2$) et les marches aléatoires paresseuses (lazy random walk).

I will present some recent progress in an ongoing project with S. Buschenhenke (Kiel), I. Ikromov (Samarkand) and D. Müller (Kiel) where we obtain the range of $p$ for which the maximal operator associated to hypersurfaces in $R^3$ is bounded on $L^p$. We will see, with a particular example, how, when the so-called height is less than $2$, it is not what determines the $p$ range.

In this talk, we will first review some classical results on the so-called ’spectral inequalities’, which yield a sharp quantification of the unique continuation of the spectral family associated with the Laplace-Beltrami operator in a compact manifold. In a second part, we will discuss how to obtain the spectral inequalities associated to the Schrodinger operator -\Delta_x + V(x), in \mathbb{R}^d, in any dimension $d\geq 1$, where V=V(x) is a real analytic potential. In particular, we can handle some long-range potentials. This is a joint work with Prof G. Lebeau (Université de Nice-Côte d'Azur, France).

Classical and flag paraproducts arise naturally in the study of nonlinear PDEs. While multi-parameter paraproduct has been studied thoroughly, known estimates for flag paraproducts only involve single parameter. We will state $L^p$ estimates for a particular case of the bi-parameter flag paraproduct on some restricted function spaces. We will discuss the key ingredient of the proof - a stopping-time argument which combines information from subspaces to obtain estimates on the entire space.

The Degasperis-Procesi equation (DP) is a spatial one-dimensional model for nonlinear shallow waters phenomena and it is one of the few known Hamiltonian PDEs which is completely integrable, namely it possesses infinitely many constants of motion. Moreover this equation is quasi-linear, namely the nonlinear terms contain derivatives of the same order of the linear part. In this talk I will show a recent result of existence and stability of small amplitude quasi-periodic solutions for Hamiltonian perturbations of the DP equation on the circle. This result is based on a combination of Nash-Moser / KAM schemes and pseudo differential calculus techniques. There are several issues in dealing with this problem:
- the equation is completely resonant, meaning that the linear solutions are all periodic, so the existence of the expected quasi-periodic solutions is due to the nonlinear terms;
- the linear dispersion is weak, in the sense that the linear solutions are close to travelling waves, and this makes difficult to impose the non-resonance Melnikov conditions required by the KAM scheme;
- the resonant structure is quite complicated and we need to exploit the integrability of the unperturbed equation to extract the first nonlinear approximate solutions from which the Nash-Moser scheme bifurcates.
This is a joint work with Roberto Feola and Michela Procesi.

Dans cet exposé, je présenterai des résultats concernant le problème de Cauchy pour les ondes non linéaires avec données aléatoires et/ou une force stochastique en dimension deux. Après avoir expliqué la construction de la mesure de Gibbs associée au Hamiltonien de l'équation et la nécessité de renormaliser, je présenterai un schéma de preuve du caractère bien posé dans le cas particulier du tore. Enfin, j'expliquerai comment contourner l'approche classique par analyse de Fourier de la construction des objets stochastiques principaux afin d'étendre ces résultats à une surface compacte sans bords plus générale.

We consider general classes of nonlinear Schr\"odinger equations on the circle with nontrivial cubic part and without external parameters. We construct a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant $M$ and a sufficiently small parameter $\varepsilon$, for generic initial data of size $\varepsilon$, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order $\varepsilon^{M+1}$. This implies that for such initial data $u(0)$ we control the Sobolev norm of the solution $u(t)$ for time of order $\varepsilon^{-M}$. Furthermore this property is locally stable: if $v(0)$ is sufficiently close to $u(0)$ (of order $\varepsilon^{3/2}$) then the solution $v(t)$ is also controled for time of order $\varepsilon^{-M}$. (Joint work with Erwan Faou and Joackim Bernier)

We consider the problem of inversion of weighted Radon transforms. This problem arises in different tomographies and, in particular, in emission tomographies. We present old and very recent results on this problem. This talk is based, in particular, on recent works [Goncharov, Novikov, 2016, 2018], [Goncharov, 2017].

We present a new proof of the dimensionless $L^p$ boundedness of the Riesz vector on manifolds with bounded geometry. The key ingredients of the proof are sparse domination and probabilistic representation of the Riesz vector. This type of proof has the significant advantage that it allows for a much stronger conclusion, giving us a new dimensionless weighted $L^p$ estimate.