We consider the conditional regularity of mild solution $${\nu}$$ ν to the incompressible Navier–Stokes equations in three dimensions. Let $${e \in \mathbb{S}^{2}}$$ e ∈ S 2 and $${0 < {T}^{*} < \infty}$$ 0 < T ∗ < ∞ . Chemin and Zhang (Ann Sci Éc Norm Supér 49:131–167, 2016) proved the regularity of $${\nu}$$ ν on (0, T*] if there exists $${p \in (4, 6)}$$ p ∈ ( 4 , 6 ) such that $$\int_{0}^{T^\ast}\|v\cdot e\|^p_{\dot{H}^{\frac{1}{2}+\frac{2}{p}}} {\rm d}t < \infty. $$ ∫ 0 T * ‖ v · e ‖ H ˙ 1 2 + 2 p p d t < ∞ . Chemin et al. (Arch Ration Mech Anal 224(3):871–905, 2017) extended the range of p to $${(4,\infty)}$$ ( 4 , ∞ ) . In this article we settle the case $${p \in [2, 4]}$$ p ∈ [ 2 , 4 ] . Our proof also works for the case $${p \in (4,\infty)}$$ p ∈ ( 4 , ∞ ) .