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

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.