  We prove an $\epsilon$-regularity criterion for the 3D Navier-Stokes
equations in terms of initial data. It shows that if a scaled local $L^2$ norm
of initial data is sufficiently small around the origin, a suitable weak
solution is regular in a set enclosed by a paraboloid started from the origin.
The result is applied to the estimate of the regular set for local energy
solutions with initial data in weighted $L^2$ spaces. We also apply this result
to studying energy concentration near a possible blow-up time and regularity of
forward discretely self-similar solutions.
