On Pluripotential Theory Associated to Quaternionic m-Subharmonic Functions
Tóm tắt
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-capacity, the relative m-extremal function, and the m-Lelong number, and show various propositions for them, based on
$$d_0$$
,
$$ d_1$$
operators, as quaternionic counterpart of
$$\partial $$
,
$$\overline{\partial }$$
, and quaternionic closed positive currents. The comparison principle and the quasicontinuity of quaternionic m-subharmonic functions are established. The definition of quaternionic m-Hessian operator can be extended to locally bounded quaternionic m-subharmonic functions and the corresponding convergence theorem is proved. We also find the fundamental solution of the quaternionic m-Hessian operator.
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