A Critical Chemin--Lerner Regularity Criterion via One Velocity Component for the Three-Dimensional Navier--Stokes Equations

We prove a scaling-critical regularity criterion involving only one velocity component for finite-energy suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations. Let $2<p<\infty$ and $m=3p/(p-2)$, so that $2/p+3/m=1$. We show that a singularity cannot occur provided