On the Evolution Kernels of Twist 2 Light-Ray Operators for Unpolarized and Polarized Deep Inelastic Scattering

The non-singlet and singlet evolution kernels of the twist--2 light-ray operators for unpolarized and polarized deep inelastic scattering are calculated in $O(\alpha_s)$ for the general case of virtualities $q^2, q'^2 \neq 0$. Special cases as the kernels for the general single-variable evolution equation and the Altarelli-Parisi and Brodsky-Lepage limits are derived from these results.