  The perturbative QCD predictions for the small $x$ behaviour of the nucleon
structure functions $F_{2,L}(x,Q^2)$ and $g_1(x,Q^2)$ are summarized. The
importance of the double logarithmic terms for the small $x$ behaviour of the
spin structure function $g_1(x,Q^2)$ is emphasised. These terms correspond to
the contributions containing the leading powers of $\alpha_s ln(1/x)^2$ at each
order of the perturbative expansion. In the non-singlet case they can be
approximately accounted for by the ladder diagrams with quark (antiquark)
exchange. We solve the corresponding integral equation with the running
coupling effects taken into account and present estimate of the effective slope
controlling the small $x$ behaviour of the non-singlet spin structure function
$g_1(x,Q^2)$ of a nucleon.
