On the regularity of the solutions of the Navier-Stokes equations via one velocity component

We consider the regularity criteria for the incompressible Navier- Stokes equations connected with one velocity component. Based on the method from Cao and Titi (2008 Indiana Univ. Math. J. 57 2643-61) we prove that the weak solution is regular, provided u(3) is an element of L-t(0, T; L-s(R-3)), 2/t + 3/s <= 3/4 + 1/2s, s > 10/3 or provided del u3 is an element of L-t (0, T; L-s(R-3)), 2/t + 3/s <= 19/12 + 1/2s if s is an element of (30/19, 3] or 2/t + 3/s <= 3/2 + 3/4s if s is an element of (3, infinity]. As a corollary, we also improve the regularity criteria expressed by the regularity of partial derivative p/partial derivative x(3) or partial derivative u(3)/partial derivative x(3).