On Kato–Ponce and fractional Leibniz

Abstract

We show that in the Kato–Ponce inequality $\Vert J^s(fg)-f{J}^{s}g{\Vert }_p\lesssim \Vert \partial f{\Vert }_{\infty }\Vert J^{s-1}g{\Vert }_p+\Vert J^{s}f{\Vert }_p\Vert g{\Vert }_{\infty }$, the $J^{s}f$ term on the right-hand side can be replaced by $J^{s-1}\partial f$. This solves a question raised in Kato–Ponce [14]. We propose a new fractional Leibniz rule for $D^s=(-\Delta {)}^{s/2}$ and similar operators, generalizing the Kenig–Ponce–Vega estimate [15] to all $s>0$. We also prove a family of generalized and refined Kato–Ponce type inequalities which include many commutator estimates as special cases. To showcase the sharpness of the estimates at various endpoint cases, we construct several counterexamples. In particular, we show that in the original Kato–Ponce inequality, the $L^{\infty }$-norm on the right-hand side cannot be replaced by the weaker BMO norm. Some divergence-free counterexamples are also included.