Octonionic Radon transform

Definition

For a compactly supported smooth function $f$ on $\mathbb O^2$ , its octonionic Radon transform is the function on affine octonionic lines defined by

(Rf)(E)=\int_E f(q)\,dq, \qquad E\in\mathcal A\mathbb OP^1,

where $dq$ is the eight-dimensional Euclidean measure induced on the affine octonionic line $E$ .

Injectivity

The transform $R$ is injective. An inversion operator averages a fourth power of the transverse eight-dimensional Laplacian over the family of octonionic lines through a point and satisfies

D(Rf)=c f

for a nonzero constant $c$ .

Role in octonionic pluripotential theory

Injectivity implies that linear combinations of distributions supported on affine octonionic lines are weakly dense among distributions on $\mathbb O^2$ . This linewise control is used to pass from restrictions of a function to positivity statements about its octonionic Hessian.

Symmetry

Both the transform and its inversion commute with translations and the $\operatorname{Spin}(9)$ action on the octonionic plane.

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: §2.