  Two main results will be presented in our paper. First, we will prove the
regularity of solutions to axially symmetric Navier-Stokes equations under a
$log$ supercritical assumption on the horizontally radial component $u^r$ and
vertical component $u^z$, accompanied by a $log$ subcritical assumption on the
horizontally angular component $u^\theta$ of the velocity. Second, the precise
Green function for the operator $-(\Delta-\frac{1}{r^2})$ under the axially
symmetric situation, where $r$ is the distance to the symmetric axis, and some
weighted $L^p$ estimates of it will be given. This will serve as a tool for the
study of axially symmetric Navier-Stokes equations. As an application, we will
prove the regularity of solutions to axially symmetric Navier-Stokes equations
under a critical (or a subcritical) assumption on the angular component
$w^\theta$ of the vorticity.
