The Radon transform on $\textrm {SL}(2,\textbf {R})/\textrm {SO}(2,\textbf {R})$
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 by D. I. Wallace and Ryuji Yamaguchi PDF
 Trans. Amer. Math. Soc. 297 (1986), 305318 Request permission
Abstract:
Let $G$ be $SL(2,{\mathbf {R}})$. $G$ acts on the upper halfplane $\mathcal {H}$ by the Möbius transformation, providing $\mathcal {H}$ with the Riemannian metric structure along with the Laplacian, $\Delta$. We study the integral transform along each geodesic. $G$ acts on $\mathcal {P}$, the space of all geodesics, in a natural way, providing $\mathcal {P}$ with its invariant measure and its own Laplacian. ($\mathcal {P}$ actually is a coset space of $G$.) Therefore the above transform can be viewed as a map from a suitable function space on $\mathcal {H}$ to a suitable function space on $\mathcal {P}$. We prove a number of properties of this transform, including the intertwining properties with its Laplacians and its relation to the Fourier transforms.References

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Additional Information
 © Copyright 1986 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 297 (1986), 305318
 MSC: Primary 22E30; Secondary 44A15, 53C65
 DOI: https://doi.org/10.1090/S00029947198608494817
 MathSciNet review: 849481