# The harmonicity of slice regular functions

## TL;DR

This paper explores harmonicity and related differential operators for slice regular functions in quaternionic analysis, establishing mean value formulas, kernel properties, and applications like Poisson and Jensen's formulas.

## Contribution

It introduces three order-two differential operators for slice regular functions, showing they are in their kernels and extending classical harmonic analysis concepts to quaternionic functions.

## Key findings

- Slice regular functions satisfy a mean value formula.
- Constructed differential operators have slice regular functions in their kernels.
- Derived Poisson and Jensen's formulas for quaternionic 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.