A remark on an endpoint Kato-Ponce inequality

TL;DR

This paper develops bilinear estimates as a step towards establishing an endpoint Kato-Ponce inequality in $L^ Infty$, including a bilinear Gagliardo-Nirenberg interpolation inequality for products of functions.

Contribution

It introduces bilinear estimates and a bilinear Gagliardo-Nirenberg inequality aimed at advancing the endpoint Kato-Ponce inequality.

Findings

Proved a bilinear Gagliardo-Nirenberg interpolation inequality.

Developed bilinear estimates towards an $L^ Infty$-endpoint Kato-Ponce inequality.

Provided theoretical foundations for future endpoint inequality proofs.

Abstract

This note introduces bilinear estimates intended as a step towards an $L^\infty$-endpoint Kato-Ponce inequality. In particular, a bilinear version of the classical Gagliardo-Nirenberg interpolation inequalities for a product of functions is proved.