Higher-order predictions for splitting functions and coefficient functions from physical evolution kernels

We have studied the physical evolution kernels for nine non-singlet observables in deep-inelastic scattering (DIS), semi-inclusive e^+e^-annihilation and the Drell-Yan (DY) process, and for the flavour-singlet case of the photon- and heavy-top Higgs-exchange structure functions (F_2, F_phi) in DIS. All known contributions to these kernels show an only single-logarithmic large-x enhancement at all powers of 1-x. Conjecturing that this behaviour persists to (all) higher orders, we have predicted the highest three (DY: two) double logarithms of the higher-order non-singlet coefficient functions and of the four-loop singlet splitting functions. The coefficient-function predictions canbe written as exponentiations of 1/N-suppressed contributions in Mellin-N space which, however, are less predictive than the well-known exponentiation of the ln^k N terms.