  This note is devoted to several inequalities deduced from a special form of
the logarithmic Hardy-Littlewood-Sobolev, which is well adapted to the
characterization of stationary solutions of a Keller-Segel system written in
self-similar variables, in case of a subcritical mass. For the corresponding
evolution problem, such functional inequalities play an important role for
identifying the rate of convergence of the solutions towards the stationary
solution with same mass.
