  We have estimated the contributions to the moments of polarized nucleon
structure function $g_1(x,Q^2)$ coming from the region of the very low x
($10^{-5}<x$). Our approach uses the nucleon structure function extrapolated to
the region of low x by the means of the double $ln^2(1/x)$ resummation. The
$Q^2$ evolution of $g_1$ was described by the unified evolution equations
incorporating both the leading order Altarelli-Parisi evolution at large and
moderate x, and the double $ln^2(1/x)$ resummation at small x. The moments were
obtained by integrating out the extrapolated nucleon structure function in the
region $10^{-5}<x<1$.
