  We study the second spatial derivatives of suitable weak solutions to the
incompressible Navier-Stokes equations in dimension three. We show that it is
locally $L ^{\frac43, q}$ for any $q > \frac43$, which improves from the
current result $L ^{\frac43, \infty}$. Similar improvements in Lorentz space
are also obtained for higher derivatives of the vorticity for smooth solutions.
We use a blow-up technique to obtain nonlinear bounds compatible with the
scaling. The local study works on the vorticity equation and uses De Giorgi
iteration. In this local study, we can obtain any regularity of the vorticity
without any a priori knowledge of the pressure. The local-to-global step uses a
recently constructed maximal function for transport equations.
