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 $${\mathbb {H}}$$ 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 $${\mathbb {H}}$$ H (analogous to an holomorphic function over $${\mathbb {C}}$$ C ) ”harmonic” in some sense, i.e., is it in the kernel of some order-two differential operator over $${\mathbb {H}}$$ H ? Finally, some applications are deduced such as a Poisson Formula for slice regular functions over $${\mathbb {H}}$$ H and a Jensen’s Formula for semi-regular ones.

On fiber bundles and quaternionic slice regular functions

“…The mapping from the finite subsets of C to the set of the monic complex polynomial Z f → f is a bijective mapping. In this sense Proposition 3.18 presents an extension of this phenomena to SRB(H) explained in terms of the fiber bundle theory and complements the results of the zero sets of slice regular functions presented in [2,15] Definition 3.20. We shall establish two fiber bundles:…”

“…Particularly, if D is a disk then the Schwarz's formula helps us to obtain f . As a consequence of the above results this work shows a simplified version of the computations presented in [19] and a supplement to [2], where the relationship between the harmonicity with the slice regularity was studied.…”

“…Paper [2] presents several relationships between the harmonic function theory with the slice regular functions. This subsection presents a consequence of the harmonicity in the interpretation of the quaternionic slice regular functions in terms of fiber bundles expanding the results obtained in [19].…”

On a Runge Theorem over $\mathbb{R}_3$

“…The theory of slice regular functions has given already many fruitful results, both on the analytic and the geometric side, see for example [3,4,6,8,9,10,11,12,13,14,16].…”

Log-biharmonicity and a Jensen formula in the space of quaternions

“…After that, in Corollaries 3.10 and 3.12, we give upper bounds on the number of zeros of a slice regular function under some additional hypotheses. The following Corollaries 3.13 and 3.14 give formulas for the computation of some integrals over H. The results contained in this paper will be further developed in the understanding of harmonic analysis on quaternionic manifolds (see [9], [13] and [12]).…”