  We revisit the dynamic spin susceptibility, $\chi(q,\omega)$, of
one-dimensional interacting fermions. To second order in the interaction,
backscattering results in a logarithmic correction to $\chi(q,\omega)$ at $q\ll
k_F$, even if the single-particle spectrum is linearized near the Fermi points.
Consequently, the dynamic spin structure factor, $\mathrm{Im}\chi(q,\omega)$,
is non-zero at frequencies above the single-particle continuum. In the boson
language, this effect results from the marginally irrelevant backscattering
operator of the sine-Gordon model. Away from the threshold, the high-frequency
tail of $\mathrm{Im}\chi(q,\omega)$ due to backscattering is larger than that
due to finite mass by a factor of $k_F/q$. We derive the renormalization group
equations for the coupling constants of the $g$-ology model at finite $\omega$
and $q$ and find the corresponding expression for $\chi(q,\omega)$, valid to
all orders in the interaction but not in the immediate vicinity of the
continuum boundary, where the finite-mass effects become dominant.
