Stability of the Faber-Krahn inequality for the short-time Fourier transform

Abstract We prove a sharp quantitative version of the Faber–Krahn inequality for the short-time Fourier transform (STFT). To do so, we consider a deficit (f;) δ (f ; Ω) which measures by how much the STFT of a function f L^2 (R) f ∈ L 2 (R) fails to be optimally concentrated on an arbitrary set R^2 Ω ⊂ R 2 of positive, finite measure. We then show that an optimal power of the deficit (f;) δ (f ; Ω) controls both the L^2 L 2 -distance of f f to an appropriate class of Gaussians and the distance of Ω to a ball, through the Fraenkel asymmetry of Ω. Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.