Lieb-Thirring inequalities and other functional inequalities for orthonormal systems

Frank, Rupert L.

doi:10.48550/arxiv.2109.13660

Cited by 1 publication

(1 citation statement)

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“…This inequality, referred to as the Lieb-Thirring inequality, was key to their proof of stability of matter; in addition to [46], we also refer the reader to [45] for further details. For wider discussion on the pursuit of obtaining versions of classical inequalities for orthonormal systems, and further examples, we direct the interested reader to [23,24,29,50,51].…”

Section: 22mentioning

confidence: 99%

On the Strichartz estimates for orthonormal systems of initial data with regularity

Bez

Hong

Lee

et al. 2019

Advances in Mathematics

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The classical Strichartz estimates for the free Schrödinger propagator have recently been substantially generalised to estimates of the formfor orthonormal systems (f j ) j of initial data in L 2 , firstly in work of Frank-Lewin-Lieb-Seiringer and later by Frank-Sabin. The primary objective is identifying the largest possible α as a function of p and q, and in contrast to the classical case, for such estimates the critical case turns out to be (p, q) = ( d+1 d , d+1 d−1 ). We consider the case of orthonormal systems (f j ) j in the homogeneous Sobolev spacesḢ s for s ∈ (0, d 2 ) and we establish the sharp value of α as a function of p, q and s, except possibly an endpoint in certain cases, at which we establish some weak-type estimates. Furthermore, at the critical case (p, q) = ( d+1 d−2s , 2s) ) for general s, we show the veracity of the desired estimates when α = p if we consider frequency localised estimates, and the failure of the (non-localised) estimates when α = p; this exhibits the difficulty of upgrading from frequency localised estimates in this context, again in contrast to the classical setting.Date: August 21, 2017. 1 A B means A ≤ CB for an appropriate constant C 1 arXiv:1708.05588v1 [math.FA]

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