On Kato-Ponce and fractional Leibniz

TL;DR

This paper refines and extends the Kato-Ponce inequality and fractional Leibniz rules, providing sharper estimates, generalizations, and counterexamples to demonstrate the limits of these inequalities.

Contribution

It replaces the $J^s f$ term with $J^{s-1} abla f$ in the Kato-Ponce inequality and introduces a new fractional Leibniz rule for $D^s$, generalizing previous estimates.

Findings

Replaced $J^s f$ with $J^{s-1} abla f$ in the inequality.

Proved a new fractional Leibniz rule for $D^s$ operators.

Constructed counterexamples showing the sharpness of the estimates.

Abstract

We show that in the Kato-Ponce inequality $\|J^s(fg)-fJ^s g\|_p \lesssim \| \partial f \|_{\infty} \| J^{s-1} g \|_p + \| J^s f \|_p \|g\|_{\infty}$, the $J^s f$ term on the RHS can be replaced by $J^{s-1} \partial f$. This solves a question raised in Kato-Ponce \cite{KP88}. We propose and prove a new fractional Leibniz rule for $D^s=(-\Delta)^{s/2}$ and similar operators, generalizing the Kenig-Ponce-Vega estimate \cite{KPV93} 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 RHS cannot be replaced by the weaker BMO norm. Some divergence-free counterexamples are also included.