  We prove a quantitative regularity theorem and blowup criterion for classical
solutions of the three-dimensional Navier-Stokes equations satisfying certain
critical conditions. The solutions we consider have
$\|r^{1-\frac3q}u\|_{L_t^\infty L_x^q}<\infty$ where $r=\sqrt{x_1^2+x_2^2}$ and
either $q\in(3,\infty)$, or $u$ is axisymmetric and $q\in(2,3]$. Using the
strategy of Tao (2019), we obtain improved subcritical estimates for such
solutions depending only on the double exponential of the critical norm. One
consequence is a double logarithmic lower bound on the blowup rate. We use some
new tools such as a decomposition of the solution that allows us to use energy
methods in these spaces, as well as a Carleman inequality for the heat equation
suited for proving quantitative backward uniqueness in cylindrical regions.
