The Faber–Krahn inequality for the short-time Fourier transform

In this paper we solve an open problem concerning the characterization of those measurable sets $\Omega\subset \mathbb{R}^{2d}$ that, among all having a prescribed Lebesgue measure, can trap largest possible energy fraction in time-frequency space, where density generic function $f\in L^2(\mathbb{R}^d)$ is defined terms its Short-time Fourier transform (STFT) $\mathcal{V} f(x,\omega)$, with Gaussian window. More precisely, given set $\Omega\subset\mathbb{R}^{2d}$ measure $s> 0$, prove that quantity \[ \Phi_\Omega=\max\Big\{\int_\Omega|\mathcal{V} f(x,\omega)|^2\,dxd\omega: f\in L^2(\mathbb{R}^d),\ \|f\|_{L^2}=1\Big\}, \] if and only $\Omega$ equivalent, up to negligible set, ball $s$, case characterize functions $f$ achieve equality. This result leads sharp uncertainty principle for "essential support" STFT (when $d=1$, be summarized by optimal bound $\Phi_\Omega\leq 1-e^{-|\Omega|}$, equality ball). Our approach, using techniques from theory after suitably rephrasing Fock also local version Lieb's inequality $L^p$ when $p\in [2,\infty)$, as well $L^p$-concentration estimates [1,\infty)$, thus proving related conjecture. cases identify corresponding extremals.