Mixed quaternionic Monge–Ampère measure

Definition

For $C^2$ quaternionic plurisubharmonic functions $u_1,\ldots,u_n$ on $\Omega\subseteq\mathbb H^n$ , their mixed quaternionic Monge–Ampère measure is

D\bigl(\operatorname{Hess}_{\mathbb H}u_1,\ldots, \operatorname{Hess}_{\mathbb H}u_n\bigr)dV,

where $D$ is the mixed discriminant of hyperhermitian matrices.

Extension and diagonal case

For continuous quaternionic PSH functions, the mixed expression extends uniquely as a nonnegative measure continuous under locally uniform convergence in each argument. On the diagonal it recovers the quaternionic Monge–Ampère measure:

\operatorname{MA}_{\mathbb H}(u) =D(\operatorname{Hess}_{\mathbb H}u,\ldots, \operatorname{Hess}_{\mathbb H}u)dV.

References

Semyon Alesker, “Valuations on convex sets, non-commutative determinants, and pluripotential theory,” Advances in Mathematics 195 (2005), 561–595. arXiv record.
Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record. Relevant: Theorem 3.6.