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 ∂ and ∂, 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 quaternionic m-subharmonic functions are established. We also find the fundamental solution of the quaternionic m-Hessian operator.