Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 024, 10 pages      arXiv:1910.12864      https://doi.org/10.3842/SIGMA.2020.024
Contribution to the Special Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry Fuchs

Horospherical Cauchy Transform on Some Pseudo-Hyperbolic Spaces

Simon Gindikin
Department of Mathematics, Hill Center, Rutgers University, 110 Frelinghysen Road, Piscataway, NJ 08854, USA

Received October 28, 2019, in final form March 29, 2020; Published online April 07, 2020

Abstract
We consider the horospherical transform and its inversion in 3 examples of hyperboloids. We want to illustrate via these examples the fact that the horospherical inversion formulas can be directly extracted from the classical Radon inversion formula. In a more broad context, this possibility reflects the fact that the harmonic analysis on symmetric spaces (Riemannian as well as pseudo-Riemannian ones) is equivalent (homologous), up to the Abelian Fourier transform, to the similar problem in the flat model. On the technical level it is important that we work not with the usual horospherical transform, but with its Cauchy modification.

Key words: pseudo-hyperbolic spaces; hyperboloids; horospheres; horospherical transform; horospherical Cauchy transform.

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References

  1. Gelfand I.M., Gindikin S.G., Graev M.I., Selected topics in integral geometry, Translations of Mathematical Monographs, Vol. 220, Amer. Math. Soc., Providence, RI, 2003.
  2. Gindikin S., Real integral geometry and complex analysis, in Integral Geometry, Radon Transforms and Complex Analysis (Venice, 1996), Lecture Notes in Math., Vol. 1684, Springer, Berlin, 1998, 70-98.
  3. Gindikin S., Integral geometry on ${\rm SL}(2;{\mathbb R})$, Math. Res. Lett. 7 (2000), 417-432.
  4. Gindikin S., Complex horospherical transform on real sphere, in Geometric Analysis of PDE and Several Complex Variables, Contemp. Math., Vol. 368, Amer. Math. Soc., Providence, RI, 2005, 227-232, arXiv:math.RT/0501023.
  5. Gindikin S., Harmonic analysis on symmetric Stein manifolds from the point of view of complex analysis, Jpn. J. Math. 1 (2006), 87-105.

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