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

TL;DR

This paper extends classical complex analysis tools like Riesz measures and Jensen formulas to the quaternionic setting, using the fundamental solution of the bilaplacian to develop new measures and formulas in four dimensions.

Contribution

It introduces a quaternionic generalization of Riesz measures and Jensen formulas, expanding the analytical framework to four-dimensional quaternionic space.

Findings

New quaternionic Riesz measures derived from the bilaplacian

Global Jensen formulas established in four-dimensional space

Extension of classical complex analysis tools to quaternionic functions

Abstract

Given a complex meromorphic function, it is well defined its Riesz measure in terms of the laplacian of the logarithm of its modulus. Moreover, related to this tool, it is possible to prove the celebrated Jensen formula. In the present paper, using among the other things the fundamental solution for the bilaplacian, we introduce a possible generalization of these two concepts in the space of quaternions, obtaining new interesting Riesz measures and global (i.e. four dimensional), Jensen formulas.