Octonionic plurisubharmonic function

Definition

Let $\Omega\subseteq\mathbb O^2$ be open. A function $u:\Omega\to[-\infty,\infty)$ is octonionic plurisubharmonic if it is upper-semicontinuous and its restriction to every affine octonionic line is subharmonic on that underlying eight-dimensional real affine space.

Smooth criterion

For $u\in C^2(\Omega)$ , octonionic plurisubharmonicity is equivalent to positive semidefiniteness of the octonionic Hessian.

Relations and closure properties

Every convex function on $\mathbb O^2\cong\mathbb R^{16}$ is octonionic PSH, and every octonionic PSH function is ordinary subharmonic. Nonnegative linear combinations and finite maxima remain octonionic PSH. The class is invariant under translations and $SL_2(\mathbb O)\cong\operatorname{Spin}(9,1)$ .

Scope

The definition belongs to the octonionic plane. Although one can write linewise conditions in other dimensions, the determinant theory needed for the established Monge–Ampère measure does not extend routinely beyond $\mathbb O^2$ .

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: §3.1.