Abstract
Abstract We establish new Strichartz estimates for orthonormal families of initial data in the case of the wave, Klein–Gordon and fractional Schrödinger equations. Our estimates extend those of Frank–Sabin in the case of the wave and Klein–Gordon equations, and generalize work of Frank et al. and Frank–Sabin for the Schrödinger equation. Due to a certain technical barrier, except for the classical Schrödinger equation, the Strichartz estimates for orthonormal families of initial data have not previously been established up to the sharp summability exponents in the full range of admissible pairs. We obtain the optimal estimates in various notable cases and improve the previous results. The main novelty of this paper is our derivation and use of estimates for weighted oscillatory integrals, which we combine with an approach due to Frank and Sabin. Our weighted oscillatory integral estimates are, in a certain sense, rather delicate endpoint versions of known dispersive estimates with power-type weights of the form $|\xi |^{-\lambda }$ or $(1 + |\xi |^2)^{-\lambda /2}$ , where $\lambda \in \mathbb {R}$ . We achieve optimal decay rates by considering such weights with appropriate $\lambda \in \mathbb {C}$ . For the wave and Klein–Gordon equations, our weighted oscillatory integral estimates are new. For the fractional Schrödinger equation, our results overlap with prior work of Kenig–Ponce–Vega in a certain regime. Our contribution to the theory of weighted oscillatory integrals has also been influenced by earlier work of Carbery–Ziesler, Cowling et al., and Sogge–Stein. Finally, we provide some applications of our new Strichartz estimates for orthonormal families of data to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces.
Highlights
For a given dispersion relation, the function denotes the solution to the initial value problem+ ( ) = 0, (0, ·) = .(, ) ∈ R1+, ≥ 1, This paper is concerned with extended versions of Strichartz estimates taking the form | |2 l (1.1)for families of orthonormal functions ( ) in a given Hilbert space H, which we shall take to be homogeneous or inhomogeneous Sobolev spaces
We provide some applications of our new Strichartz estimates for orthonormal families of data to the theory of infinite systems of Hartree type, weighted velocity averaging lemmas for kinetic transport equations, and refined Strichartz estimates for data in Besov spaces
We present our weighted oscillatory integral estimates corresponding to the wave equation, the Klein–Gordon equation and the fractional Schrödinger equation, which we need for proof of the orthonormal Strichartz estimates
Summary
For a given dispersion relation , the function denotes the solution to the initial value problem. For families of orthonormal functions ( ) in a given Hilbert space H, which we shall take to be homogeneous or inhomogeneous Sobolev spaces This particular line of investigation originated in recent work of Frank et al [29] for the Schrödinger propagator ( ( ) = | |2). The idea to generalize classical inequalities from a single-function input to an orthonormal family traces back further, with pioneering work of Lieb–Thirring [57] establishing extended versions of certain Gagliardo–Nirenberg– Sobolev inequalities and applications to the stability of matter. We provide substantial progress in this direction, generalizing work in [29] to the fractional Schrödinger case and significantly extending the results in [30] for the wave and Klein–Gordon equations. To facilitate the presentation of prior results and our new results in the extended framework (1.1), we first review the classical (single-function) case
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