The Kato-Ponce Inequality

TL;DR

This paper revisits the classical Kato-Ponce inequality, extending its validity to quasi-Banach spaces with restrictions, and introduces a multi-parameter version for partial fractional derivatives.

Contribution

It provides a simplified proof of the Kato-Ponce inequality, extends its applicability to quasi-Banach spaces, and develops a multi-parameter variant for partial derivatives.

Findings

Validity of the inequality in quasi-Banach spaces with restrictions

Counter-example demonstrating sharpness of restrictions

Multi-parameter variant for partial fractional derivatives

Abstract

In this article we develop a simplistic approach to revisit the classical Kato-Ponce inequality, which is also known as 'fractional Leibniz rule.' As a consequence, we derive the validity of this inequality even in quasi-Banach spaces $L^p$ for $p<1$ with a certain restrictions on the indices. Also, we display the sharpness of this restriction by means of a counter-example. Finally, we also prove a multi-parameter variant of the inequality, which allows for partial fractional derivatives on $R^n$.