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The k-plane transform on the Heisenberg group

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The paper deals with the k-plane transform \({\mathcal {R}}_{k}\) on the Heisenberg group. We study the properties of the transform \({\mathcal {R}}_{k}\) and obtain three types of inversion formulas for \({\mathcal {R}}_{k}\). The first inversion is deduced with the help of the group Fourier transform, together with the partial Riesz potential and Heisenberg sublaplacian. By this formula, another two formulas are established in terms with the adjoint of \({\mathcal {R}}_{k}\) and the wavelet, respectively.

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  1. 1.

    Aldrovandi, R., Pereira, J.G.: An Introduction to Geometrical Physics. World Scientific Publishing Co., Inc, River Edge (1995)

  2. 2.

    Felix, R.: Radon-transformation auf nilpotenten Lie-Gruppen. (German). Invent. Math. 112, 413–443 (1993)

  3. 3.

    Geller, D., Stein, E.M.: Singular convolution operators on the Heisenberg group. Bull. Am. Math. Soc. 6, 99–103 (1982)

  4. 4.

    Geller, D., Stein, E.M.: Estimates for singular convolution operators on the Heisenberg group. Math. Ann. 267, 1–15 (1984)

  5. 5.

    Gonzalez, F.B.: Notes on integral geometry and harmonic analysis. COE Lecture Note, 24. Math-for-Industry (MI) Lecture Note Series. Kyushu University, Faculty of Mathematics, Fukuoka (2010)

  6. 6.

    He, J.: An inversion formula of the Radon transform on the Heisenberg group. Can. Math. Bull. 47, 389–397 (2004)

  7. 7.

    He, J., Liu, H.: Inversion of the Radon transofrm associated with the classical domain of type one. Int. J. Math. 16, 875–887 (2005)

  8. 8.

    He, J., Liu, H.: Admissible wavelets and inverse Radon transform assiciated with the affine homogeneous Siegel domains of type II. Commun. Anal. Geom. 15, 1–28 (2007)

  9. 9.

    He, J., Xiao, J.: Inversion of the Radon transform on the free nilpotent Lie group of step two. Can. J. Math. 66, 700–720 (2014)

  10. 10.

    Helgason, S.: Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions. Pure and Applied Mathematics, vol. 113. Academic Press Inc, Orlando (1984)

  11. 11.

    Helgason, S.: Integral Geometry and Radon transforms. Springer, New York (2011)

  12. 12.

    Keinert, F.: Inversion of \(k\)-plane transforms and applications in computer tomography. SIAM Rev. 31, 273–298 (1989)

  13. 13.

    Leahy, J.V., Smith, K.T., Solmon, D.C.: Uniqueness, nonuniqueness and inversion in the X-ray and Radon problems. CINESTAV-IPN (1992)

  14. 14.

    Nessibi, M.M., Trimèche, K.: An inversion formula of the Radon transform on the Laguerre hypergroup by using generalized wavelets. J. Math. Anal. Appl. 208, 337–363 (1997)

  15. 15.

    Peng, L., Zhang, G.: Radon transform on H-type and Siegel-type nilpotent groups. Int. J. Math. 18, 1061–1070 (2007)

  16. 16.

    Qu, G.: Wavelet inversion of the \(k\)-plane transform and its application. Appl. Comput. Harmon. Anal. 21, 262–267 (2006)

  17. 17.

    Rubin, B.: The Calderón formula, windowed \(X\)-ray transforms and Radon transforms in \(L^p\) spaces. J. Fourier Anal. Appl. 4, 175–197 (1998)

  18. 18.

    Rubin, B.: Inversion of \(k\)-plane transforms via continuous wavelet transforms. J. Math. Anal. Appl. 220, 187–203 (1998)

  19. 19.

    Rubin, B.: Convolution-backprojection method for the \(k\)-plane transform, and Calderón’s identity for ridgelet transforms. Appl. Comput. Harmon. Anal. 16, 231–242 (2004)

  20. 20.

    Rubin, B.: The Radon transform on the Heisenberg group and the transversal Radon transform. J. Funct. Anal. 262, 234–272 (2012)

  21. 21.

    Rubin, B.: Introduction to Radon Transforms. With Elements of Fractional Calculus and Harmonic Analysis. Encyclopedia of Mathematics and Its Applications, vol. 160. Cambridge University Press, New York (2015)

  22. 22.

    Solmon, D.C.: The X-ray transform. J. Math. Anal. Appl. 56, 61–83 (1976)

  23. 23.

    Solmon, D.C.: A note on \(k\)-plane integral transforms. J. Math. Anal. Appl. 71, 351–358 (1979)

  24. 24.

    Strichartz, R.S.: \(L^{p}\) estimates for Radon transforms in Euclidean and non-Euclidean spaces. Duke Math. J. 48, 699–727 (1981)

  25. 25.

    Strichartz, R.S.: \(L^{p}\) harmonic analysis and Radon transforms on the Heisenberg group. J. Funct. Anal. 96, 350–406 (1991)

  26. 26.

    Thangavelu, S.: An Introduction to the Uncertainty Principle: Hardy’s Theorem on Lie Groups. Progress in Mathematics, vol. 217. Birkhäuser Boston, Inc., Boston (2004)

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The work for this paper is supported by the National Natural Science Foundation of China (Nos. 11501131, 11671414) and Guangdong Basic and Applied Basic Research Foundation (2019A1515010955).

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Correspondence to Jianxun He.

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Xiao, J., He, J. The k-plane transform on the Heisenberg group. J. Pseudo-Differ. Oper. Appl. 11, 289–309 (2020). https://doi.org/10.1007/s11868-019-00314-1

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  • k-plane transform
  • Radon transform
  • Heisenberg group
  • Inversion formula
  • Riesz potential

Mathematics Subject Classification

  • 43A85
  • 44A12