I will start improving the definition of slice regular function over the quaternions given by Gentili-Struppa in 2006-2007.
Then, bringing new ideas to the theory, I will answer positively to the question: is a slice regular function over the quaternions (analogously to a holomorphic function over the complex) ”harmonic” in some sense, i.e. is it in the kernel of some order-two differential operator over the quaternions?
Finally, I will deduce novel integral formulas as applications.
This is part of a common project with J. Winkelmann.