Quaternionic Monge–Ampère measure

Definition

For a $C^2$ quaternionic plurisubharmonic function $u$ on $\Omega\subseteq\mathbb H^n$ , its quaternionic Monge–Ampère measure is

\operatorname{MA}_{\mathbb H}(u) =\det_M\!\left(\frac{\partial^2u} {\partial\bar q_i\,\partial q_j}\right)dV,

where $\det_M$ is the Moore determinant and $dV$ is Lebesgue measure.

Continuous potentials

For a continuous quaternionic PSH function, this expression has a unique extension as a nonnegative Borel measure characterized by agreement with the smooth formula and continuity under locally uniform convergence. This is the quaternionic analogue of Aleksandrov's real Hessian measure and the Chern–Levine–Nirenberg/Bedford–Taylor complex Monge–Ampère measure.

Scope

Continuity of $u$ is part of this basic definition. More singular quaternionic PSH functions require additional pluripotential-theoretic domains of definition; one should not apply the determinant measure to an arbitrary unbounded function without specifying such a class.

References

Semyon Alesker, “Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables,” Bulletin des Sciences Mathématiques 127 (2003), 1–35. arXiv record. Relevant: Theorem 3.4.
Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record. Relevant: §3.