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 \(\delta (f;\Omega )\) which measures by how much the STFT of a function \(f\in L^{2}(\mathbb{R})\) fails to be optimally concentrated on an arbitrary set \(\Omega \subset \mathbb{R}^{2}\) of positive, finite measure. We then show that an optimal power of the deficit \(\delta (f;\Omega )\) controls both the \(L^{2}\)-distance of \(f\) to an appropriate class of Gaussians and the distance of \(\Omega \) to a ball, through the Fraenkel asymmetry of \(\Omega \). Our proof is completely quantitative and hence all constants are explicit. We also establish suitable generalizations of this result in the higher-dimensional context.