  This paper is devoted mainly to the global existence problem for the
two-dimensional parabolic-parabolic Keller-Segel in the full space. We derive a
critical mass threshold below which global existence is ensured. Using
carefully energy methods and ad hoc functional inequalities we improve and
extend previous results in this direction. The given threshold is supposed to
be the optimal criterion, but this question is still open. This global
existence result is accompanied by a detailed discussion on the duality between
the Onofri and the logarithmic Hardy-Littlewood-Sobolev inequalities that
underlie the following approach.
