  We study the regular sets of local energy solutions to the Navier-Stokes
equations in terms of conditions on the initial data. It is shown that if a
weighted $L^2$ norm of the initial data is finite, then all local energy
solutions are regular in a region confined by space-time hypersurfaces
determined by the weight. This result refines and generalizes Theorems C and D
of Caffarelli, Kohn and Nirenberg (Comm. Pure Appl. Math. 35; 1982) and our
recent paper [15] as well.
