Well-posedness for the Navier-Stokes equations with data in homogeneous Sobolev-Lorentz spaces

In this paper, we study local well-posedness for the Navier-Stokes equations (NSE) with arbitrary initial data in homogeneous Sobolev-Lorentz spaces ḢLq,rs(Rd):=(−Δ)−s/2Lq,r for d≥2,q>1,s≥0, 1≤r≤∞, and dq−1≤sd,r=q,s=0 (see Cannone (1995), Cannone and Meyer (1995)), for q=r=2,d2−1