On quaternionic pluripotential theory associated to quaternionic $m$-subharmonic functions

TL;DR

This paper extends pluripotential theory to quaternionic $m$-subharmonic functions, introducing new operators, measures, and principles, and establishing foundational results like convergence, comparison, and fundamental solutions.

Contribution

It introduces quaternionic versions of key pluripotential concepts, extending the theory to quaternionic $m$-subharmonic functions with new operators and fundamental solutions.

Findings

Defined quaternionic $m$-Hessian operator and measures

Proved convergence and comparison principles

Established fundamental solutions for the quaternionic $m$-Hessian operator

Abstract

Many aspects of pluripotential theory are generalized to quaternionic $m$-subharmonic functions. We introduce quaternionic version of notions of the $m$-Hessian operator, $m$-subharmonic functions, $m$-Hessian measure, $m$-capapcity, the relative $m$-extremal function and the $m$-Lelong number, and show various propositions for them, based on $d_0$ and $ d_1$ operators, the quaternionic counterpart of $\partial$ and $\overline{\partial}$, and quaternionic closed positve currents. The definition of quaternionic $m$-Hessian operator can be extended to locally bounded quaternionic $m$-subharmonic functions and the corresponding convergence theorem is proved. The comparison principle and the quasicontinuity of bounded quaternionic $m$-subharmonic functions are established. We also find the fundamental solution of the quaternionic $m$-Hessian operator.