  The structure function $g_1(x,Q^2)$ is analysed within the formalism based on
unintegrated spin dependent parton distributions incorporating the LO
Altarelli-Parisi evolution and the double $ln^2(1/x)$ resummation at low $x$.
We quantitatively examine possible role of the latter for the small $x$
behaviour of the structure functions $g_1$ of the nucleon within the region
which may be relevant for the possible polarized HERA measurements. We show
that while the non-singlet structure function is dominated at low $x$ by ladder
diagrams the contribution of the non-ladder bremsstrahlung terms is important
for the singlet structure function. Predictions for the polarized gluon
distribution $\Delta G(x,Q^2)$ at low $x$ are also given.
