  We consider a family of weights which permit to generalize the Leray
procedure to obtain weak suitable solutions of the 3D incom-pressible
Navier-Stokes equations with initial data in weighted L 2 spaces. Our principal
result concerns the existence of regular global solutions when the initial
velocity is an axisymmetric vector field without swirl such that both the
initial velocity and its vorticity belong to L 2 ((1 + r 2) -- $\gamma$ 2 dx),
with r = x 2 1 + x 2 2 and $\gamma$ $\in$ (0, 2).
