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We study the large-x behaviour of the physical evolution kernels for flavournon-singlet observables in deep-inelastic scattering, where x is the Bjorkenvariable, semi-inclusive e^+ e^- annihilation and Drell-Yan lepton-pairproduction. Unlike the corresponding MSbar-scheme coefficient functions, allthese kernels show a single-logarithmic large-x enhancement at all orders in1-x. We conjecture that this universal behaviour, established byFeynman-diagram calculations up to the fourth order, holds at all orders in thestrong coupling constant alpha_s. The resulting predictions are presented forthe highest ln^n(1-x) contributions to the higher-order coefficient functions.In Mellin-N space these predictions take the form of an exponentiation which,however, appears to be less powerful than the well-known soft-gluonexponentiation of the leading (1-x)^(-1) ln^n(1-x) terms. In particular indeep-inelastic scattering the 1/N corrections are non-negligible for allpractically relevant N.

On non-singlet physical evolution kernels and large-x coefficient functions in perturbative QCD