A resummation of large sub-leading corrections at small x.

The NLL corrections to the BFKL kernel are known to be very large, to the extent that even for small values of alpha(s), they lead to physical cross sections which are not positive definite. It is shown in the context of a toy model, that such pathological behaviour is an artifact of the truncation at NLL order, and is associated in particular with double transverse logarithms. These are resummed in a manner consistent with the full NLL kernel, and are shown to change its properties quite considerably.