  It is well understood that the leading logarithmic approximation for the
amplitudes of high energy processes is insufficient and that the
next-to-leading logarithmic effects are very large and lead to instability of
the solution. The resummation at low $x$, which includes kinematical
constraints and other corrections leads to stable result. Using previously
established resummation procedure we study in detail the preasymptotic effects
which occur in the solution to the resummed BFKL equation when the energy is
not very large. We find that in addition to the well known reduction of the
intercept, which governs the energy dependence of the gluon Green's function,
resummation leads to the delay of the onset of its small $x$ growth. Moreover
the gluon Green's function develops a dip or a plateau in wide range of
rapidities, which increases for large scales. The preasymptotic region in the
gluon Green's function extends to about $8$ units in rapidity for the
transverse scales of the order of $30-100 \; {\rm GeV} $. To visualize the
expected behavior of physical processes with two equal hard scales we calculate
the cross section of the process $\gamma^{*}+\gamma^{*}\to X$ to be probed at
future very high-energy electron-positron colliders. We find that at
$\gamma^*\gamma^*$ energies below $100 \; \rm GeV$ the BFKL Pomeron leads to
smaller value of the cross section than the Born approximation, and only starts
to dominate at energies about $100 \; \rm GeV$. This pattern is significantly
different from the one which we find using LL approximation. We also analyze
the transverse momentum contributions to the cross section for different
virtualities of the photons and find that the dominant contributions to the
integral over the transverse momenta comes from lower values than the the
external scales in the process under consideration.
