The Harmonicity of Slice Regular Functions

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.