The orthonormal Strichartz inequality on torus

Nakamura, Shohei

doi:10.1090/tran/7982

Cited by 11 publications

(6 citation statements)

References 45 publications

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“…which is valid for all ∈ R and > 0 (see, for example, [77, p. 205]). We also remark that the analogous estimate to (7.14) on the torus was used in [62] to establish local well-posedness of infinite systems of Hartree equations in the periodic setting.…”

Section: Proof (Proof Ofmentioning

confidence: 91%

Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

Bez

Lee

Nakamura

2021

Forum of Mathematics, Sigma

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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.

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Section: Proof (Proof Ofmentioning

confidence: 91%

Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

Bez

Lee

Nakamura

2021

Forum of Mathematics, Sigma

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“…For further results and open problems, see [10,152,11,12]. For applications of these bounds to the dynamics of quantum many-body systems, see, for instance, [121,122].…”

Section: Magnetic Lieb-thirring Inequalities the Lieb-thirring Inequa...mentioning

confidence: 99%

The Lieb–Thirring inequalities: Recent results and open problems

Frank¹

2021

Nine Mathematical Challenges

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This review celebrates the generous gift by Ronald and Maxine Linde for the remodeling of the Caltech mathematics department and the author is very grateful to the editors of this volume for the invitation to contribute. We attempt to survey recent results and open problems connected to Lieb-Thirring inequalities. In view of several excellent existing reviews [132,17,113,91,108] as well as highly recommended textbooks [134,136], we sometimes put our focus on developments during the past decade.The author would like to thank all his collaborators on the topic of Lieb-Thirring inequalities and, in particular, A.

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“…The space-time exponent of this generalization is different from estimate (29) and the initial data is measured in Schatten norm. We refer to [6,14,16,27] for interested readers.…”

Section: Remarkmentioning

confidence: 99%

“…Proposition 1 Let γ (t, x, y) = e −it(H x − Hy ) γ 0 (x, y) be the solution to Eq. (27), then for any T > 0 and s ≥ 0,…”

Section: Strichartz and Collapsing Estimatesmentioning

confidence: 99%

The Hartree equation with a constant magnetic field: well-posedness theory

Dong

2021

Lett Math Phys

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We consider the Hartree equation for infinitely many electrons with a constant external magnetic field. For the system, we show a local well-posedness result when the initial data is the pertubation of a Fermi sea, which is a non-trace class stationary solution to the system. In this case, the one particle Hamiltonian is the Pauli operator, which possesses distinct properties from the Laplace operator, for example, it has a discrete spectrum and infinite-dimensional eigenspaces. The new ingredient is that we use the Fourier-Wigner transform and the asymptotic properties of associated Laguerre polynomials to derive a collapsing estimate, by which we establish the local wellposedness result.

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The orthonormal Strichartz inequality on torus

Cited by 11 publications

References 45 publications

Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

Strichartz estimates for orthonormal families of initial data and weighted oscillatory integral estimates

The Lieb–Thirring inequalities: Recent results and open problems

The Hartree equation with a constant magnetic field: well-posedness theory

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