Octonionic Monge–Ampère measure

Definition

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

\operatorname{MA}_{\mathbb O}(u) =\det\!\left(\operatorname{Hess}_{\mathbb O}u\right)dV,

using the quadratic determinant on $H_2(\mathbb O)$ .

Continuous potentials

For continuous octonionic PSH functions, the smooth expression extends uniquely to a nonnegative Borel measure such that locally uniform convergence of potentials implies weak convergence of measures. This is the octonionic analogue of the Aleksandrov and Chern–Levine–Nirenberg continuity theorems.

Max-min identity

If $u,v$ and $\min(u,v)$ are continuous octonionic PSH functions, then the corresponding determinant measures satisfy the same inclusion–exclusion identity under $\max$ and $\min$ as in the quaternionic Błocki formula. This identity is what turns support-function Hessian measures into valuations.

Scope

The measure is defined here on $\mathbb O^2$ . It should not be presented as an octonionic Monge–Ampère operator in arbitrary dimension.

References

Semyon Alesker, “Plurisubharmonic functions on the octonionic plane and $\operatorname{Spin}(9)$ -invariant valuations on convex sets,” Journal of Geometric Analysis 18 (2008), 651–686. arXiv record. Relevant: §§3.2–3.3.