On the Critical One Components Regularity for the $3-D$ Navier-Stokes System in $L_T^p({\dot{B}}_{2,\mathrm{\infty }}^{\frac{1}{2}+\frac{2}{p}})$ spaces

We consider the conditional regularity of the mild solution $v$ of the $3-D$ incompressible Navier-Stokes equations with initial data $v_0\in {\dot{H}}^{\frac{1}{2}}$ and vorticity ${\mathrm{\Omega }}_0\in L^{r_0}$ for some $r_0\in (1,2)$. We prove that if the solution associated with initial data $v_0$ blows up at a finite time $T^{\ast }$, then for any $2<p<\mathrm{\infty }$, and any unit vectors $e$ in ${\mathbb{R}}^3$, the integral $${\int }_0^{T^{\ast }}{\parallel (v(t)\mathrm{\mid }e{)}_{{\mathbb{R}}^3}\parallel }_{{\dot{B}}_{2,\mathrm{\infty }}^{\frac{1}{2}+\frac{2}{p}}}^p\mathrm{d}t$$ blows up at $T^{\ast }$. The conclusion improves the recent results in Chemin et al. (Arch Ration Mech Anal 224(3):871-905, 2017) and Han et al. (Arch. Rational Mech. Anal. 231:939-970, 2019).