  We show how perturbation theory may be reorganized to give splitting
functions which include order by order convergent sums of all leading
logarithms of $x$. This gives a leading twist evolution equation for parton
distributions which sums all leading logarithms of $x$ and $Q^2$, allowing
stable perturbative evolution down to arbitrarily small values of $x$.
Perturbative evolution then generates the double scaling rise of $F_2$ observed
at HERA, while in the formal limit $x\to 0$ at fixed $Q^2$ the Lipatov
$x^{-\lambda}$ behaviour is eventually reproduced. We are thus able to explain
why leading order perturbation theory works so well in the HERA region.
