The Faber–Krahn inequality for the short-time Fourier transform

In this paper we solve an open problem concerning the characterization of those measurable sets Ω ⊂ R2d that, among all sets having a prescribed Lebesgue measure, can trap the largest possible energy fraction in time-frequency space, where the energy density of a generic function f∈ L2(Rd) is defined in terms of its Short-time Fourier transform (STFT) Vf(x, ω) , with Gaussian window. More precisely, given a measurable set Ω ⊂ R2d having measure s> 0 , we prove that the quantity Φ_Ω=max{∫_Ω|Vf(x,ω)|2dxdω:f∈L2(Rd),‖f‖L2=1} is largest possible if and only if Ω is equivalent, up to a negligible set, to a ball of measure s, and in this case we characterize all functions f that achieve equality. This result leads to a sharp uncertainty principle for the “essential support” of the STFT (when d= 1, this can be summarized by the optimal bound Φ Ω≤ 1 - e^-| Ω |, with equality if and only if Ω is a ball). Our approach, using techniques from measure theory after suitably rephrasing the problem in the Fock space, also leads to a local version of Lieb’s uncertainty inequality for the STFT in Lp when p∈ [2 , ∞) , as well as to Lp-concentration estimates when p∈ [1 , ∞) , thus proving a related conjecture. In all cases we identify the corresponding extremals.