Bernstein inequalities via the heat semigroup

We extend the classical Bernstein inequality to a general setting including Schr{\"o}dinger operators and divergence form elliptic operators on Riemannian manifolds or domains. Moreover , we prove a new reverse inequality that can be seen as the dual of the Bernstein inequality. The heat kernel will be the backbone of our approach but we also develop new techniques such as semi-classical Bernstein inequalities, weak factorization of smooth functions {\`a} la Dixmier-Malliavin and BM O -- L $\infty$ multiplier results (in contrast to the usual L $\infty$ -- BM O ones). Also, our approach reveals a link between the L p-Bernstein inequality and the boundedness on L p of the Riesz transform. The later being an important subject in harmonic analysis. 2010 Mathematics Subject Classifications: 35P20, 58J50, 42B37 and 47F05.Comment: Revised version, to appear in Math. An

NORMAL FORMS FOR SEMILINEAR QUANTUM HARMONIC OSCILLATORS

International audienceWe consider the semilinear harmonic oscillator i\psi_t=(-\Delta +\va{x}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d,\ t\in \R where $M$ is a Hermite multiplier and $g$ a smooth function globally of order 3 at least.\\ We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on $M$ related to the non resonance of the linear part, this normal form is integrable when $d=1$ and gives rise to simple (in particular bounded) dynamics when $d\geq 2$.\\ As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions

Normal Forms for Semilinear Quantum Harmonic Oscillators

We consider the semilinear harmonic oscillator i\psi_t=(-\Delta +\va{x}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \R^d, t\in \R where $M$ is a Hermite multiplier and $g$ a smooth function globally of order 3 at least. We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on $M$ related to the non resonance of the linear part, this normal form is integrable when $d=1$ and gives rise to simple (in particular bounded) dynamics when $d\geq 2$. As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions

Long time existence for the semi-linear beam equation on irrational tori of dimension two

We prove a long time existence result for the semi-linear beam equation with small and smooth initial data. We use a regularizing effect of the structure of beam equations and a very weak separation property of the spectrum of an irrational torus under a Diophantine assumption on the radius. Our approach is inspired from a paper by Zhang about the Klein-Gordon equation with a quadratic potential

A NECESSARY AND SUFFICIENT CONDITION FOR PROBABILISTIC CONTINUITY ON A BOUNDARYLESS COMPACT RIEMANNIAN MANIFOLD

We give a necessary and sufficient condition for the uniform convergence of random series of eigenfunctions on a boundaryless compact Riemannian manifold. Due to the lack of homogeneity of a compact manifold (by comparison with the case of compact groups studied by Marcus and Pisier), our proof relies on a suitable generalization of the Dudley-Fernique obtained via the theory of majorizing measures. As a consequence, we generalize an estimate of Burq and Lebeau about the supremum of a random eigenfunction. Finally, we prove that our results are universal w.r.t. the random variables (thus generalizing a result of Marcus and Pisier), w.r.t. compact submanifolds and w.r.t. the Riemannian structure of the underlying manifold

ETUDE DYNAMIQUE DE QUELQUES EQUATIONS AUX DERIVEES PARTIELLES HAMILTONIENNES NON LINEAIRES A POTENTIEL CONFINANT

This thesis is concerned by stability of solutions of some non linear Schroedinger partial differential equations (PDE) on Rn with a confining potential and a regular initial condition.Two potentials are studied : the harmonic oscillator multidimensional and the polynomial confining potential unidimensional.In our context, the stability means roughly the following : the solution exists on a time-interval whose length depends polynomially on the smallness of the initial condition (almost global existence) and stays near the solution of an explicit completely integrable equation with the same initial condition.We use the Birkhoff's normal forms theory to handle our issue.The key point is the Hamiltonian structure of our PDE.We create an abstract differential model (which encompasses our PDE) and prove that it has a Birkhoff's normal form of all order, ie a proper renormalization of the Hamiltonian which ensures in particular the stability.L'objet de la thèse est l'étude de la stabilité des solutions de certaines équations aux dérivées partielles (EDP) non linéaires de type Schrödinger à potentiel confinant sur Rn et à condition initiale régulière.Les deux potentiels étudiés sont l'oscillateur harmonique multi-dimensionnel et le potentiel polynomial confinant unidimensionnel. La stabilité en question peut se résumer ainsi : il existe un intervalle temporel d'existence de la solution telle que sa longueur dépend de façon polynomiale de la petitesse de la condition initiale (existence presque globale) et sur lequel la solution reste dynamiquement proche de la solution d'une équation explicite complètement intégrable (avec même condition initiale).Nous utilisons la théorie des formes normales de Birkhoff pour aborder notre problème.Le point clé est le caractère hamiltonien des EDP concernées.Nous créons un modèle différentiel abstrait (qui comprend l'EDP étudiée) et l'on y démontre l'existence de formes normales de Birkhoff à tout ordre, c'est-à-dire des renormalisations adéquates de l'hamiltonien qui en l'occurrence impliquent la stabilité

Normal form for semi-linear Klein–Gordon equations with superquadratic oscillators

Nous prouvons un résultat d'existence, sur des grands temps, pour des équations semi-linéaires de Klein-Gordon avec un potentiel surquadratique pour des petites conditions initiales régulières. La preuve repose sur une faible séparation des valeurs propres et des estimations spécifiques des modes propres des oscillateurs surquadratiques.We prove a long time existence result for semi-linear Klein-Gordon equations with a superquadratic potential for small and smooth initial datum. The proof relies on a weak separation of eigenvalues and specific multilinear estimates of eigenfunctions of superquadratic oscillators