Cegrell classes and a variational approach for the quaternionic Monge-Ampere equation

TL;DR

This paper develops a variational framework for solving quaternionic Monge-Ampere equations by introducing Cegrell classes of quaternionic plurisubharmonic functions, extending the operator's domain, and establishing key properties like integration by parts.

Contribution

It introduces finite energy Cegrell classes for quaternionic plurisubharmonic functions and applies a variational approach to solve quaternionic Monge-Ampere equations on hyperconvex domains.

Findings

Extended the quaternionic Monge-Ampere operator to new classes.

Proved integration by parts and comparison principles in these classes.

Solved quaternionic Monge-Ampere equations with finite energy measures.

Abstract

In this paper, we introduce finite energy classes of quaternionic plurisubharmonic functions of Cegrell type and study the quaternionic Monge-Ampere operator on these classes on quaternionic hyperconvex domains of Hn. We extend the domain of definition of quaternionic Monge-Ampere operator to some Cegrell classes, the functions of which are not necessarily bounded. We show that integration by parts and comparison principle are valid on some classes. Moreover, we use the variational method to solve the quaternionic Monge-Ampere equations when the right hand side is a positive mea- sure of finite energy.