The harmonicity of slice regular functions

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.

The exposition of this paper has been improved a lot following the valuable suggestions of a careful Referee that we warmly thank. The paper will appear soon on The Journal of Geometric Analysis

References in corpus (10)

• The Schwarz-Pick lemma for slice regular functions
• Singularities of slice regular functions over real alternative *-algebras
• Slice regularity and harmonicity on Clifford algebras
• Regular vs. classical Möbius transformations of the quaternionic unit ball
• Slice-quaternionic Hopf surfaces
• On a quaternionic Picard theorem
• Landau's theorem for slice regular functions on the quaternionic unit ball
• On Quaternionic Tori and their Moduli Spaces
• Log-biharmonicity and a Jensen formula in the space of quaternions
• On Runge Pairs and topology of axially symmetric domains