Kato-Ponce Inequality for Mixed Local and Nonlocal Differential Operators

We establish Kato-Ponce inequalities for the mixed local and nonlocal differential operator Formula: see text and its inhomogeneous variant for a full range of Lebesgue indices including the endpoint case, and determine the sharp constraint condition of the regularity index. The homogeneous inequality applies over the same range as in the classical scenario, whereas the inhomogeneous inequality covers a broader range of indices. Furthermore, we derive sharp Kato-Ponce commutator estimates within this framework. Our proof is based on an adaption of Fourier analytic methods developed by Grafakos-Oh 36, Bourgain-Li 9 and Oh-Wu 54 to the present setting, combined with a new lower bound estimate for the family of operators Formula: see text.