A variational approach to the quaternionic Hessian equation

TL;DR

This paper develops a variational method to solve quaternionic m-Hessian equations by introducing finite energy classes of quaternionic m-plurisubharmonic functions and defining the associated operator on these classes.

Contribution

It introduces finite energy classes of quaternionic m-plurisubharmonic functions and applies a variational approach to solve the quaternionic m-Hessian equation for positive Radon measures.

Findings

Defined quaternionic m-Hessian operator on Cegrell's classes.

Established existence of solutions for the quaternionic m-Hessian equation.

Extended variational methods to quaternionic analysis.

Abstract

In this paper, we introduce finite energy classes of quaternionic $m$-plurisubharmonic functions of Cegrell type and define the quaternionic $m$-Hessian operator on some Cegrell's classes. We use the variational approach to solve the quaternionic $m$-Hessian equation when the right-hand side is a positive Radon measure.