Contents Online
Mathematical Research Letters
Volume 27 (2020)
Number 4
On $L^1$ endpoint Kato–Ponce inequality
Pages: 1129 – 1163
DOI: https://dx.doi.org/10.4310/MRL.2020.v27.n4.a8
Authors
Abstract
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