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 $\delta (f;\mathrm{\Omega })$ $\delta (f;\Omega )$ which measures by how much the STFT of a function $f\in L^2(\mathbb{R})$ $f\in L^{2}(\mathbb{R})$ fails to be optimally concentrated on an arbitrary set $\mathrm{\Omega }\subset {\mathbb{R}}^2$ $\Omega \subset \mathbb{R}^{2}$ of positive, finite measure. We then show that an optimal power of the deficit $\delta (f;\mathrm{\Omega })$ $\delta (f;\Omega )$ controls both the $L^2$ $L^{2}$ -distance of $f$ $f$ to an appropriate class of Gaussians and the distance of $\mathrm{\Omega }$ $\Omega $ to a ball, through the Fraenkel asymmetry of $\mathrm{\Omega }$ $\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.