Littlewood-Paley decomposition of operator densities and application to a new proof of the Lieb-Thirring inequality

The goal of this note is to prove a analogue of the Littewood-Paley decomposition for densities of operators and to use it in the context of Lieb-Thirring inequalities

The Hartree equation for infinitely many particles. II. Dispersion and scattering in 2D

We consider the nonlinear Hartree equation for an interacting gas containing infinitely many particles and we investigate the large-time stability of the stationary states of the form $f(-\Delta)$, describing an homogeneous Fermi gas. Under suitable assumptions on the interaction potential and on the momentum distribution $f$, we prove that the stationary state is asymptotically stable in dimension 2. More precisely, for any initial datum which is a small perturbation of $f(-\Delta)$ in a Schatten space, the system weakly converges to the stationary state for large times

Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces

We generalize the $L^p$ spectral cluster bounds of Sogge for the Laplace-Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. The optimality of these new bounds is also discussed. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators.Comment: 30 page

The Stein-Tomas inequality in trace ideals

The goal of this review is to explain some recent results regarding generalizations of the Stein-Tomas (and Strichartz) inequalities to the context of trace ideals (Schatten spaces).Comment: Proceedings of the Laurent Schwartz semina

Maximizers for the Stein-Tomas inequality

We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein-Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein-Tomas inequality. Our result is valid in any dimension.Comment: 37 page