  We show that if a Leray-Hopf solution $u$ to the 3D Navier-Stokes equation
belongs to $C((0,T]; B^{-1}_{\infty,\infty})$ or its jumps in the
$B^{-1}_{\infty,\infty}$-norm do not exceed a constant multiple of viscosity,
then $u$ is regular on $(0,T]$. Our method uses frequency local estimates on
the nonlinear term, and yields an extension of the classical
Ladyzhenskaya-Prodi-Serrin criterion.
