The Harmonicity of Slice Regular Functions
The Journal of Geometric Analysis(2020)
摘要
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 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 ℍ (analogous to an holomorphic function over ℂ ) ”harmonic” in some sense, i.e., is it in the kernel of some order-two differential operator over ℍ ? Finally, some applications are deduced such as a Poisson Formula for slice regular functions over ℍ and a Jensen’s Formula for semi-regular ones.
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关键词
Slice regular functions,Harmonicity,Laplacians,Mean value theorems,Quaternionic analysis,Poisson formula,Jensen formula
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