On the critical one component regularity for 3-D Navier-Stokes system: general case

Let us consider an initial data $v_0$ for the homogeneous incompressible 3D Navier-Stokes equation with vorticity belonging to $L^{\frac{3}{2}}\cap L^2$. We prove that if the solution associated with $v_0$ blows up at a finite time $T^{\star }$, then for any $p$ in $]4,\mathrm{\infty }[$, and any unit vector $e$ of ${\mathbb{R}}^3$, the $L^p$ norm in time with value in ${\dot{H}}^{\frac{1}{2}+\frac{2}{p}}$ of $(v\mathrm{\mid }e{)}_{{\mathbb{R}}^3}$ blows up at $T^{\star }$