  We deal with a weak solution v to the Navier-Stokes initial value problem in
R^3 x(0,T). We denote by \omega^+ a spectral projection of \omega=\curl\, v,
defined by means of the spectral resolution of identity associated with the
self-adjoint operator \curl. We show that certain conditions imposed on
\omega^+ or, alternatively, only on \omega^+_3 (the third component of
\omega^+) imply regularity of solution v.
