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Mathematical Physics
Title: On a variational problem related to the Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities
(Submitted on 7 Mar 2024)
Abstract: We explicitly solve a variational problem related to upper bounds on the optimal constants in the Cwikel--Lieb--Rozenblum (CLR) and Lieb--Thirring (LT) inequalities, which has recently been derived in [Invent. Math. 231 (2023), no.1, 111-167. this https URL ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. this https URL ]. We achieve this through a variational characterization of the $L^1$ norm of the Fourier transform of a function and duality, from which we obtain a reformulation in terms of a variant of the Hadamard three lines lemma. By studying Hardy-like spaces of holomorphic functions in a strip in the complex plane, we are able to provide an analytic formula for the minimizers, and use it to get the best possible upper bounds for the optimal constants in the CLR and LT inequalities achievable by the method of [Invent. Math. 231 (2023), no.1, 111-167. this https URL ] and [J. Eur. Math. Soc. (JEMS) 23 (2021), no.8, 2583-2600. this https URL ].
Submission history
From: Thiago Carvalho Corso [view email][v1] Thu, 7 Mar 2024 09:21:39 GMT (38kb,D)
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