In , we proved that any transformation which preserves solutions of the wave equation is a similarity, an inversion, or a Bateman transformation. This paper gives a generalization of the results to ultra-hyperbolic type equations.
Let Ω be a bounded domain of RN (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 ∈L1loc(Ω) such that ∆u ∈ L1loc(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.