On an endpoint Kato-Ponce inequality

We prove that the $L^{\infty}$ end-point Kato-Ponce inequality (Leibniz rule) holds for the fractional Laplacian operators $D^s=(-\Delta)^{s/2}$, $J^s=(1-\Delta)^{s/2}$, $s>0$. This settles a conjecture by Grafakos, Maldonado and Naibo [7]. We also establish a family of new refined Kato-Ponce commutator estimates. Some of these inequalities are in borderline spaces.