On the uniqueness and the regularity of solutions to the Navier Stokes in the Lorentz space $L^{3,\infty}(\mathbb{R}^3)$

We prove uniqueness of solutions to the Navier-Stokes equations in the space \[ L^p\bigl([0,T],\widetilde{L}^{3,\infty}(\mathbb{R}^3)\bigr) \cap C\bigl([0,T],B^{-1,\infty}_{\infty}(\mathbb{R}^3)\bigr), \qquad p>2, \] where $\widetilde{L}^{3,\infty}(\mathbb{R}^3)$ denotes the closure of the Schwartz