[en] We determine the asymptotic behavior of the deep inelastic leptoproduction structure functions in the quasielastic region z -> 1 in the leading double logarithmic approximation of QCD. It is shown how the asymptotic behavior is related to the ON/OFF limit of the Sudakov form factor. The running coupling constant is incorporated by comparing the double logarithmic approximation with the solution of the renormalization group equation. We compare our result with various suggestions to improve the convergence of the perturbative expansion in the quasielastic region. (orig.)