Complete O($α$) QED corrections to the process $ep \to eX$ in mixed variables

The complete set of ${\cal O}(\alpha)$ QED corrections with soft photon exponentiation to the process $ep \rightarrow eX$ in mixed variables ($y=y_h,Q^2=Q_l^2$) is calculated in the quark parton model, including the lepton-quark interference and the quarkonic corrections which were unknown so far. The interference corrections amount to few percent or less and become negligible at small $x$. The leading logarithmic terms proportional to $\ln(Q^2/m_q^2)$ from radiation off quarks are discussed and the non-logarithmic quarkonic corrections found to be negligible for almost all experimentally accessible $x$ and $y$.