We improve regularity criteria for weak solutions to the Navier-Stokes equations stated in references [1], [3] and [12], by using in the proof given in [3], a new idea introduced by H. O. Bae and H. J. Choe in [1]. This idea allows us, in one of the main hypothesis (see eq. (1.7)), to replace the velocity u by its projection \( \bar u \) into an arbitrary hyperplane of \( {\Bbb R}^n \); see Theorem A. For simplicity, we state our results for space dimension \( n \le 4 \), since if \( n \ge 5 \) the proofs become more technical and additional hypotheses are needed. However, for the interested reader, we will present the formal calculations for arbitrary dimension n.