Journal of Inequalities and Applications
Volume 6 (2001), Issue 1, Pages 29-36
doi:10.1155/S1025583401000030
Some remarks on Kato’s inequality
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.
Abstract
Let N≥1 and N>1. Let Ω be a domain of ℝN. In this article we shall establish Kato’s inequalities for p-harmonic operators Lp. Here Lp is defined as Lpu=div(|∇u|p−2∇u) for u∈Kp(Ω), where Kp(Ω) is an admissible class. If p=2 for example, then we have Kp(Ω)={u∈Lloc1(Ω):∂ju,∂j,k2u∈Lloc1(Ω) for j,k=1,2,…,N}. Then we shall prove that Lp|u|≥(sgnu)Lpu and Lpu+≥(sgn+u)p−1Lpu in 𝒟′(Ω) with u∈Kp(Ω). These inequalities are called Kato’s inequalities provided that p=2.