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 which measures by how much the STFT of a function fails to be optimally concentrated on an arbitrary set of positive, finite measure. We then show that an optimal power of the deficit controls both the -distance of to an appropriate class of Gaussians and the distance of to a ball, through the Fraenkel asymmetry of . Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.