Mathematical Research Letters

Volume 27 (2020)

Number 4

On $L^1$ endpoint Kato–Ponce inequality

Pages: 1129 – 1163



Seungly Oh (Department of Mathematics, Western New England University, Springfield, Massachusetts, U.S.A.)

Xinfeng Wu (Department of Mathematics, China University of Mining & Technology, Beijing, China)


We prove that the following endpoint Kato–Ponce inequality holds:\[{\lVert D^s (fg) \rVert}_{L^{\frac{q}{q+1}} (\mathbb{R}^n)}\lesssim{\lVert D^s f \rVert}_{L^1 (\mathbb{R}^n)}{\lVert g \rVert}_{L^q (\mathbb{R}^n)} \\+{\lVert f \rVert}_{L^1 (\mathbb{R}^n)}{\lVert D^s g \rVert}_{L^q (\mathbb{R}^n)}\; \textrm{,}\]for all $1 \leq q \leq \infty$, provided $s \gt n/q$ or $s \in 2 \mathbb{N}$. Endpoint estimates for several variants of Kato–Ponce inequality in mixed norm Lebesgue spaces are also presented. Our results complement and improve some existing results.

Received 22 March 2019

Accepted 22 March 2019

Published 14 December 2020