Monday, April 15 at 4:15 p.m. (Santiago time) in the seminar room, Beauchef 851, 5th floor.
Speaker: Tobias Ried (TU Munich)
Title: On a variational problem related to the Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities.
Abstract: In this talk I will introduce a variational problem arising in the derivation of upper bounds on the optimal constants in the Cwikel-Lieb-Rozenblum (CLR) inequality based upon a substantial refinement of Cwikel’s original proof. The approach we developed with D. Hundertmark, P. Kunstmann and S. Vugalter in [Invent. Math. 231 (2023), no.1, 111-167] highlights a natural but overlooked connection of optimal bounds on the CLR constant with bounds for maximal Fourier multipliers from harmonic analysis.
I will show how, through a variational characterization of the L1 norm of the Fourier transform of a function and convex duality, this variational problem can be reformulated in terms of a variant of the classical Hadamard three lines lemma. By studying Hardy-like spaces of holomorphic functions in a strip in the complex plane, together with T. Carvalho Corso, we were able to derive 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 Hundertmark-Kunstmann-Vugalter and myself.