In [4], 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.

We study star and semistar operations on 1-dimensional Prüfer domains D. We prove a uniqueness theorem for y-representations of sigma operations on D.

In this paper we introduce the notion of presemistar operation and enlarge the theory of semistar operation to the theory of presemistar operation.

Let Ω be a bounded domain of R^N (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 ∈ L¹_loc(Ω) such that ∆u ∈ L¹_loc(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.