  Let $v$ be the velocity of Leray-Hopf solutions to the axially symmetric
three-dimensional Navier-Stokes equations. Under suitable conditions for
initial values, we prove the following a priori bound \[ |v(x, t)| \le
\frac{C}{r^2} |\ln r|^{1/2}, \]where $r \in (0, 1/2)$ is the distance from $x$
to the z axis, and $C$ is a constant depending only on the initial value.
  This provides a pointwise upper bound (worst case scenario) for possible
singularities while the recent papers \cite{CSTY2} and \cite{KNSS} gave a lower
bound. The gap is polynomial order 1 modulo a half log term.
