  We derive a lower bound for the Wehrl entropy in the setting of SU(1,1). For
asymptotically high values of the quantum number k, this bound coincides with
the analogue of the Lieb-Wehrl conjecture for SU(1,1) coherent states. The
bound on the entropy is proved via a sharp norm bound. The norm bound is
deduced by using an interesting identity for Fisher information of SU(1,1)
coherent state transforms and a new family of sharp Sobolev inequalities on the
hyperbolic plane. To prove the sharpness of our Sobolev inequality, we need to
first prove a uniqueness theorem for solutions of a semi-linear Poisson
equation (which is actually the Euler-Lagrange equation for the variational
problem associated with our sharp Sobolev inequality) on the hyperbolic plane.
Uniqueness theorems proved for similar semi-linear equations in the past do not
apply here and the new features of our proof are of independent interest, as
are some of the consequences we derive from the new family of Sobolev
inequalities.
