  Axially symmetric solutions to the Navier-Stokes equations in a bounded
cylinder are considered. On the boundary the normal component of the velocity
and he angular components of the velocity and vorticity are assumed to vanish.
If the norm of the initial swirl is sufficiently small, then the regularity of
axially symmetric, weak solutions is shown. The key tool is a new estimate for
the stream function in certain weighted Sobolev spaces.
