Regularity of a Weak Solution to the Navier-Stokes Equations via One Component of a Spectral Projection of Vorticity

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.