A Proof of Lieb--Wehrl Entropy conjecture for $SU(N,1)$

Abstract: We investigate the sharp functional inequalities for the coherent state transforms of $SU(N,1)$. These inequalities are rooted in Wehrl's definition of semiclassical entropy and his conjecture about its minimum value. Lieb resolved this conjecture in 1978, posing a similar question for Bloch coherent states of $SU(2)$. The $SU(2)$ conjecture was settled by Lieb and Solovej in 2014, and the conjecture was extended for a wide class of Lie groups. The generalized Lieb conjecture has been resolved for several Lie groups, including $SU(N),\, N\geq2$, $SU(1,1)$, and its $AX+B$ subgroup. With sharp functional inequalities for the coherent state transforms of the group $SU(N,1)$, we confirm this Lieb-Wehrl entropy conjecture for $SU(N,1),\, N\geq2$. Additionally, we explore the Faber-Krahn inequality, which applies to the short-time Fourier transform with a Gaussian window. This inequality was previously proven by Nicola and Tilli and later extended by Ramos and Tilli to the wavelet transform. In this paper, we further extend this result within the framework of the Bergman space $\mathcal A_{\alpha}$.