Journal of Inequalities and Applications
Volume 6 (2001), Issue 1, Pages 29-36

Some remarks on Kato’s inequality

Toshio Horiuchi

Department of Mathematical Sciences, Ibaraki University, Mito, Ibaraki 310, Japan

Received 30 August 1999; Revised 4 November 1999

Copyright © 2001 Toshio Horiuchi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let N1 and N>1. Let Ω be a domain of N. In this article we shall establish Kato’s inequalities for p-harmonic operators Lp. Here Lp is defined as Lpu=div(|u|p2u) for uKp(Ω), where Kp(Ω) is an admissible class. If p=2 for example, then we have Kp(Ω)={uLloc1(Ω):ju,j,k2uLloc1(Ω) for j,k=1,2,,N}. Then we shall prove that Lp|u|(sgnu)Lpu and Lpu+(sgn+u)p1Lpu in 𝒟(Ω) with uKp(Ω). These inequalities are called Kato’s inequalities provided that p=2.