  In a series of recent papers we extended the notion of square integrability,
for representations of nilpotent Lie groups, to that of stepwise square
integrability. There we discussed a number of applications based on the fact
that nilradicals of minimal parabolic subgroups of real reductive Lie groups
are stepwise square integrable. Here, in Part I, we prove stepwise square
integrability for nilradicals of arbitrary parabolic subgroups of real
reductive Lie groups. This is technically more delicate than the case of
minimal parabolics. We further discuss applications to Plancherel formulae and
Fourier inversion formulae for maximal exponential solvable subgroups of
parabolics and maximal amenable subgroups of real reductive Lie groups.
Finally, in Part II, we extend a number of those results to (infinite
dimensional) direct limit parabolics. These extensions involve an infinite
dimensional version of the Peter-Weyl Theorem, construction of a direct limit
Schwartz space, and realization of that Schwartz space as a dense subspace of
the corresponding $L^2$ space.
