Quaternionic plurisubharmonic function

Definition

Let $\Omega\subseteq\mathbb H^n$ be open. A function $u:\Omega\to[-\infty,\infty)$ is quaternionic plurisubharmonic if it is upper-semicontinuous and, for every affine right quaternionic line $L=q_0+v\mathbb H$ , the restriction $u|_{\Omega\cap L}$ is subharmonic on the underlying real four-dimensional line, or identically $-\infty$ on a component.

Smooth criterion

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

\left(\frac{\partial^2u}{\partial\bar q_i\,\partial q_j}\right).

This is the quaternionic counterpart of positivity of the real Hessian for a convex function and of the Levi form for a complex PSH function.

Relation to convexity and subharmonicity

Every convex function on a convex open subset of $\mathbb H^n\cong \mathbb R^{4n}$ is quaternionic PSH, and every quaternionic PSH function is ordinary subharmonic. The first inclusion is generally strict. These are genuine quaternionic analogues, not an identification of convex, complex PSH, and quaternionic PSH functions on a common domain.

Basic closure properties

Finite maxima and nonnegative linear combinations of quaternionic PSH functions remain quaternionic PSH. Locally uniform limits do as well, provided the limit is not identically $-\infty$ on a component. For continuous quaternionic PSH functions, the quaternionic Monge–Ampère measure extends the smooth Hessian determinant.

References

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