Nonlocal generalization of the axial anomaly and $x$ dependence of the anomalous gluon contribution

The generalization of the axial anomaly is considered. It is shown that bilocal axial quark operators on the light cone possess, in addition to the pointlike anomaly, also a lightlike anomaly. The consequences for the definition of anomaly-free quark distribution functions and the effect of both the gluon coefficient function and splitting kernels in polarized deep inelastic scattering are discussed.