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The continuity and range of the quaternionic Monge–Ampère operator on quaternionic space

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Abstract

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 \((\triangle u_j)^n\rightarrow (\triangle u)^n\). We also obtain an equivalent condition of “convergence in \(C_{n-1}\)-capacity” by using methods from Xing (Proc Am Math Soc 124(2):457–467, 1996). As an application, the range of the quaternionic Monge–Ampère operator is discussed.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Shenzhen University, Room 415, Science and Technology Building, Shenzhen, 518060, People’s Republic of China

    Dongrui Wan

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  1. Dongrui Wan
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Correspondence to Dongrui Wan.

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This work is supported by National Nature Science Foundation in China (No. 11401390, 11571305).

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Wan, D. The continuity and range of the quaternionic Monge–Ampère operator on quaternionic space. Math. Z. 285, 461–478 (2017). https://doi.org/10.1007/s00209-016-1716-8

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  • DOI: https://doi.org/10.1007/s00209-016-1716-8

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