We generalize the theorems of Stein-Tomas and Strichartz about surface restrictions of Fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions. We deduce the corresponding Strichartz bounds for solutions to Schrödinger equations up to the endpoint, thereby solving an open problem of Frank, Lewin, Lieb and Seiringer. We also prove uniform Sobolev estimates in Schatten spaces, extending the results of Kenig, Ruiz, and Sogge. We finally provide applications of these results to a Limiting Absorption Principle in Schatten spaces, to the well-posedness of the Hartree equation in Schatten spaces, to Lieb-Thirring bounds for eigenvalues of Schrödinger operators with complex potentials, and to Schatten properties of the scattering matrix.

Abstract. 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 (−∆), 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 (−∆) in a Schatten space, the system weakly converges to the stationary state for large times.

We show local and global well-posedness results for the Hartree equationwhere γ is a bounded self-adjoint operator on L 2 (R d ), ρ γ (x) = γ (x, x) and w is a smooth short-range interaction potential. The initial datum γ (0) is assumed to be a perturbation of a translation-invariant state γ f = f (− ) which describes a quantum system with an infinite number of particles, such as the Fermi sea at zero temperature, or the Fermi-Dirac and Bose-Einstein gases at positive temperature. Global well-posedness follows from the conservation of the relative (free) energy of the state γ (t), counted relatively to the stationary state γ f . We indeed use a general notion of relative entropy, which allows us to treat a wide class of stationary states f (− ). Our results are based on a Lieb-Thirring inequality at positive density and on a recent Strichartz inequality for orthonormal functions, which are both due to Frank, Lieb, Seiringer and the first author of this article.

Abstract. We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein-Tomas inequality. In particular, if a wellknown 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. Main resultA fundamental result in harmonic analysis is the Stein-Tomas theorem [30,36], which states that if f ∈ L 2 (S N −1 ), N ≥ 2, then the inverse Fourier transformf of f dω, with dω the surface measure on S N −1 , that is, and its L q (R N ) norm is bounded by a constant times the L 2 (S N −1 ) norm of f . Moreover, it is well known that the exponent q is optimal (smallest possible) for this to hold for any f ∈ L 2 (S N −1 ). In this paper we are interested in the optimal Stein-Tomas constant,where · denotes the norm in L 2 (S N −1 ). The value of R N and optimizing functions are only known in dimension N = 3 due to a remarkable work of Foschi [15]; see [11] for partial progress in N = 2. Our main concern here is whether the supremum defining R N is attained and, more generally, the description of maximizing sequences for R N . These questions were recently considered in fundamental papers by Christ and Shao, where the existence of a maximizer for N = 3 [12] and N = 2 [29] was shown, as well as a precompactness result for maximizing sequences for N = 3 [13]. What makes dimensions N = 2 and 3 special is that the exponent q in (1.1) is an even integer, so that one can multiply out |f | q . Our results will be valid in any dimension.c 2016 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes. Christ and Shao discovered that for the problem of existence of an maximizer for R N a key role is played by the Strichartz inequality [32]. The optimal constant in this inequality isThe overall factor (2π) −(d+2)/d and the factor 1/2 in front of the Laplacian are not important, but simplify some formulas below. We say that a sequence (The following is our main result.up to modulations and, in particular, there is a maximizer for R N .Clearly, the optimization problem for R N is invariant under modulations, so precompactness up to modulations is the best one can expect. Our theorem says that assumption (1.2) is sufficient for this. In fact, it is easy to see that (1.2) is also necessary for the precompactness modulo modulations of all maximizing sequences. We will comment on this in Remark 2.5, where we will also see that (1.2) holds with ≥ instead of >.As we will argue below, in dimensions N = 2 and N = 3, the strict inequality (1.2) holds and so we recover the Christ-Shao results on the existence of optimizers [12,29] and precompactness in N = 3 [13] and we obtain, for the first time, precompactness of maximizing sequences for N = 2.We believe, but cannot prove, that the strict inequality (1.2) holds in any dimension. To verify it, it seems natural to first compute S N −1 and then to use a perturbation argument to establish (1.2). In ...