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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.doi:10.1088/1126-6708/1998/07/019 fatcat:u3xsngowbrc4hh4bcfs5ayg3ky