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

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 δ (f;Ω ) δ ( f ; Ω ) which measures by how much the STFT of a function f∈ L²(R) f ∈ L 2 ( R ) fails to be optimally concentrated on an arbitrary set Ω ⊂ R² Ω ⊂ R 2 of positive, finite measure. We then show that an optimal power of the deficit δ (f;Ω ) δ ( f ; Ω ) controls both the L² L 2 -distance of f f to an appropriate class of Gaussians and the distance of Ω Ω to a ball, through the Fraenkel asymmetry of Ω 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.