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.

中文翻译：

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在$ L ^ 1 $端点上，Kato-Ponce不等式

我们证明以下端点Kato–Ponce不等式成立：\ [{\ lVert D ^ s（fg）\ rVert} _ {L ^ {\ frac {q} {q + 1}}（\ mathbb {R} ^ n ）} \ lesssim {\ lVert D ^ sf \ rVert} _ {L ^ 1（\ mathbb {R} ^ n）} {\ lVert g \ rVert} _ {L ^ q（\ mathbb {R} ^ n）} \\ + {\ lVert f \ rVert} _ {L ^ 1（\ mathbb {R} ^ n）} {\ lVert D ^ sg \ rVert} _ {L ^ q（\ mathbb {R} ^ n）} \\ ; \ textrm {，} \]对于所有$ 1 \ leq q \ leq \ infty $，只要提供$ s \ gt n / q $或$ s \ in 2 \ mathbb {N} $。还提供了混合范数Lebesgue空间中Kato-Ponce不等式的几种变体的端点估计。我们的结果补充并改进了一些现有结果。