Sharp one component regularity for Navier-Stokes equations

    We consider the conditional regularity of mild solution $v$ to the incompressible Navier-Stokes equations in three dimensions. Let $e \in \mathbb{S}^2$ and $0 < t^\ast < \infty$. j. chemin and p. zhang (ann. sci. \'{e}c. norm. sup\'{e}r, 2016 ) proved the regularity of $v$ on $(0,t^\ast]$ if there exists $p \in (4, 6)$ such that $$\int_0^{t^\ast}\|v\cdot e\|^p_{\dot{h}^{\frac{1}{2}+\frac{2}{p}}}dt < \infty.$$ j. chemin, p. zhang and z. f. zhang  (arch. ration. mech. ana. , 2017) extended the range of $p$ to $(4, \infty)$. in this article we settle the case $p \in [2, 4]$.  our proof also works for the case $p \in (4,\infty)$.