  We obtain a Beale-Kato-Majda-type criterion with optimal frequency and
temporal localization for the 3D Navier-Stokes equations. Compared to previous
results our condition only requires the control of Fourier modes below a
critical frequency, whose value is explicit in terms of time scales. As
applications it yields a strongly frequency-localized condition for regularity
in the space $B^{-1}_{\infty,\infty}$ and also a lower bound on the decaying
rate of $L^p$ norms $2\leq p <3$ for possible blowup solutions. The proof
relies on new estimates for the cutoff dissipation and energy at small time
scales which might be of independent interest.
