Abstract
In this article, we investigate harmonicity, Laplacians, mean value theorems, and related topics in the context of quaternionic analysis. We observe that a Mean Value Formula for slice regular functions holds true and it is a consequence of the well-known Representation Formula for slice regular functions over {mathbb {H}}. Motivated by this observation, we have constructed three order-two differential operators in the kernel of which slice regular functions are, answering positively to the question: is a slice regular function over {mathbb {H}} (analogous to an holomorphic function over {mathbb {C}}) ”harmonic” in some sense, i.e., is it in the kernel of some order-two differential operator over {mathbb {H}}? Finally, some applications are deduced such as a Poisson Formula for slice regular functions over {mathbb {H}} and a Jensen’s Formula for semi-regular ones.
Highlights
In [20] and [21], Gentili and Struppa gave the following definition of slice regular function over the quaternions: Definition 1.1 Let be a domain in H
We show that for any point p ∈ H and every 3-sphere S containing p in the interior, there exists a H-valued measure on S such that f ( p) = S f (q)dμ(q) for every slice regular function f (Theorem 7.1)
Over the field of complex numbers, the mean value property is equivalent to harmonicity. It is natural ask ourselves if slice regular functions were in the kernel of some order-two differential operator over H: in Sect. 8 we answer positively to this question constructing three order-two differential operators in the kernel of which slice regular functions are
Summary
In [20] and [21], Gentili and Struppa gave the following definition of slice regular function over the quaternions: Definition 1.1 Let be a domain in H. We show that for any point p ∈ H and every 3-sphere S containing p in the interior, there exists a H-valued measure on S such that f ( p) = S f (q)dμ(q) for every slice regular function f (Theorem 7.1). Over the field of complex numbers, the mean value property is equivalent to harmonicity It is natural ask ourselves if slice regular functions were in the kernel of some order-two differential operator over H: in Sect. The first one is ∗, introduced in Definition 6.15 For slice functions it is defined everywhere, for other functions only outside R. We hope that this paper can provide new ideas for studying slice regular functions and their “harmonic properties” on slice regular quaternionic manifolds recently introduced by Bisi-Gentili in [12] and Angella-Bisi in [5]
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