Siegel-Radon transforms of transverse dynamical systems

TL;DR

This paper generalizes the Radon transform to dynamical systems, providing criteria for boundedness and applications to embedding representations, with explicit formulas for special cases like cut-and-project sets and the Heisenberg group.

Contribution

It introduces a broad framework for Radon transforms in dynamical systems, extending classical definitions and applying to various geometric and analytical contexts.

Findings

Established criteria for boundedness, integrability, and square-integrability of Siegel-Radon transforms.

Derived explicit formulas for dual transforms in cut-and-project sets.

Embedded Schr"odinger representations into $L^2$-spaces using aperiodic Zak transforms in the Heisenberg group.

Abstract

We extend Helgason's classical definition of a generalized Radon transform, defined for a pair of homogeneous spaces of an lcsc group $G$, to a broader setting in which one of the spaces is replaced by a possibly non-homogeneous dynamical system over $G$ together with a suitable cross section. This general framework encompasses many examples studied in the literature, including Siegel (or $\Theta$-) transforms and Marklof-Str\"ombergsson transforms in the geometry of numbers, Siegel-sVeech transforms for translation surfaces, and Zak transforms in time-frequency analysis. Our main applications concern dynamical systems $(X, \mu)$ in which the cross section is induced from a separated cross section. We establish criteria for the boundedness, integrability, and square-integrability of the associated Siegel-Radon transforms, and show how these transforms can be used to embed induced $G$-representations into $L^p(X, \mu)$ for appropriate values of $p$. These results apply in particular to hulls of approximate lattices and certain "thinnings" thereof, including arbitrary positive density subsets in the amenable case. In the special case of cut-and-project sets, we derive explicit formulas for the dual transforms, and in the special case of the Heisenberg group we provide isometric embedding of Schr\"odinger representations into the $L^2$-space of the hulls of positive density subsets of approximate lattices in the Heisenberg group by means of aperiodic Zak transforms.