What means "slightly supercritical" in Navier-Stokes equations?

The goal of this paper is to clarify some differences between the critical Lebesgue space $L^3$ and the critical weak Lebesgue space $L^{3,\mathrm{\infty }}$, when these are considered in the hypothesis of classical statements for the 3D homogeneous incompressible Navier-Stokes equations.