Kato-Ponce inequality for fractional nonlocal parabolic operators

We establish Kato-Ponce inequality (or fractional Leibniz rule) for fractional nonlocal parabolic operators ${({\partial }_t-{△}_x)}^{s/2}$ and ${(1+{\partial }_t-{△}_x)}^{s/2}$ of arbitrary order $s>0$ for a full range of Lebesgue indices including the endpoints, and determine the sharp range of s. We also prove a sharp Kato-Ponce commutator inequality for ${(1+{\partial }_t-{△}_x)}^{s/2}$. To achieve these results, we not only adapt the methods of Bourgain-Li [11], Grafakos-Oh [25] and Oh-Wu [50] to the present parabolic setting, but build up sharp decay estimates for higher-order hyper-singular integrals of Nogin-Rubin [48] and Stinga-Torrea [54], which are crucial for us to derive the sharp ranges of s.