In this paper, we introduce the first-order differential operators d0 and d1 acting on the quaternionic version of differential forms on the flat quaternionic space H n . The behavior of d0, d1 and △ = d0d1 is very similar to ∂, ∂ and ∂∂ in several complex variables. The quaternionic Monge-Ampère operator can be defined as (△u) n and has a simple explicit expression. We define the notion of closed positive currents in the quaternionic case, and extend several results in complex pluripotential theory to the quaternionic case: define the Lelong number for closed positive currents, obtain the quaternionic version of Lelong-Jensen type formula, and generalize Bedford-Taylor theory, i.e., extend the definition of the quaternionic Monge-Ampère operator to locally bounded quaternionic plurisubharmonic functions and prove the corresponding convergence theorem.

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In this paper, we use the quaternionic closed positive currents to establish some pluripotential results for quaternionic Monge-Ampère operator. By introducing a new quaternionic capacity, we prove a sufficient condition which implies the weak convergence of quaternionic Monge-Ampère measures ( u j ) n → ( u) n . We also obtain an equivalent condition of "convergence in C n−1 -capacity" by using methods from Xing (Proc Am Math Soc 124 (2): [457][458][459][460][461][462][463][464][465][466][467] 1996). As an application, the range of the quaternionic Monge-Ampère operator is discussed.

Quaternionic Monge-Ampère equations have recently been studied intensively using methods from pluripotential theory. We present an alternative approach by using the viscosity methods. We study the viscosity solutions to the Dirichlet problem for quaternionic Monge-Ampère equationsHere Ω is a bounded domain on the quaternionic space H n , g ∈ C(∂Ω), and F (q, t) is a continuous function on Ω × R → R + which is non-decreasing in the second variable. We prove a viscosity comparison principle and a solvability theorem. Moreover, the equivalence between viscosity and pluripotential solutions is showed.

m-subharmonic functions are the right class of admissible solutions to the complex Hessian equation. In this paper, we generalize the definition of the complex Hessian operator to some unbounded m-subharmonic functions, and we prove that the complex Hessian operator is continuous on the monotonically decreasing sequences of m-subharmonic functions. Moreover we establish the Lelong-Jensen type formula and introduce the Lelong number for m-subharmonic functions. A useful inequality for the mixed Hessian operator is showed.