Remarks on Kato's inequality when ∆pu is a measure

Abstract

Let Ω be a bounded domain of R (N ≥ 1) . In this article, we shall study Kato’s inequality when ∆pu is a measure, where ∆pu denotes a p-Laplace operator with 1 < p < ∞. The classical Kato’s inequality for a Laplacian asserts that given any function u ∈ Lloc(Ω) such that ∆u ∈ Lloc(Ω), then ∆(u) is a Radon measure and the following holds: ∆(u) ≥ χ[u≥0]∆u in D′(Ω). Our main result extends Kato’s inequality to the case where ∆pu is a Radon measures on Ω. We also establish the inverse maximum principle for ∆p.