Differential and Integral Equations

On an endpoint Kato-Ponce inequality

Jean Bourgain and Dong Li

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We prove that the $L^{\infty}$ end-point Kato-Ponce inequality (Leibniz rule) holds for the fractional Laplacian operators $D^s=(-\Delta)^{s/2}$, $J^s=(1-\Delta)^{s/2}$, $s>0$. This settles a conjecture by Grafakos, Maldonado and Naibo [7]. We also establish a family of new refined Kato-Ponce commutator estimates. Some of these inequalities are in borderline spaces.

Article information

Differential Integral Equations, Volume 27, Number 11/12 (2014), 1037-1072.

First available in Project Euclid: 18 August 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25: Maximal functions, Littlewood-Paley theory 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems


Bourgain, Jean; Li, Dong. On an endpoint Kato-Ponce inequality. Differential Integral Equations 27 (2014), no. 11/12, 1037--1072. https://projecteuclid.org/euclid.die/1408366784

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