Skip to main content
SpringerLink
Log in

End point estimates for Radon transform of radial functions on non-Euclidean spaces

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We prove end point estimate for Radon transform of radial functions on affine Grasamannian and real hyperbolic space. We also discuss analogs of these results on the sphere.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anker, J.P.: Sharp estimates for some functions of the laplacian on noncompact symmetric spaces. Duke Math. J. 65(2), 257–297 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Anker, J.P.: The spherical fourier transform of rapidly decreasing functions. a simple proof of a characterization due to harish-chandra, helgason, trombi, and varadarajan. J. Funct. Anal. 96(2), 331–349 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Berenstein, C.A., Rubin, B.: Radon transform of \(l^p\)-functions on the lobachevsky space and hyperbolic wavelet transforms. Forum Math. 5(11), 567–590 (1999)

    MathSciNet  Google Scholar 

  4. Berenstein, C.A., Rubin, B.: Totally Geodesic Radon Transform of \(L^p\)-Functions on Real Hyperbolic Space. Fourier Analysis and Convexity, pp. 37–58. Applied Numerical Harmonical Analysis, Birkhuser Boston, Boston, MA (2004)

  5. Bray, W. O.: Aspects of harmonic analysis on real hyperbolic space. In: Fourier analysis (Orono, ME, 1992). Lecture Notes in Pure and Applied Mathematics, vol. 157, pp. 77–102. Dekker, New York (1994)

  6. Cowling, M., Meda, S., Setti, A.G.: An overview of harmonic analysis on the group of isometries of a homogeneous tree. Exposition. Math. 16(5), 385–423 (1998)

    MATH  MathSciNet  Google Scholar 

  7. Duoandikoetxea, J., Naibo, V., Oruetxebarria, O.: \(k\)-plane transforms and related operators on radial functions. Michigan Math. J. 49, 265–276 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, Inc., NJ (2004)

    MATH  Google Scholar 

  9. Gonzalez, F.B.: Radon transform on grassmann manifolds. J. Funct. Anal. 71, 339–362 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gonzalez, F.B., Kakehi, T.: Pfaffian systems and radon transforms on affine grassmann manifolds. Math. Ann. 326(2), 237–273 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gonzalez, F.B., Kakehi, T.: Dual radon transforms on affine grassmann manifolds. Trans. Am. Math. Soc. 356(10), 4161–4180 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Helgason, S.: The radon transform on euclidean spaces, compact two-point homogeneous spaces and grassmann manifolds. Acta Math. 113, 153–180 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  13. Helgason, S.: The Radon transform, 2nd edn. Progress in Mathematics, vol. 5. Birkhuser Boston Inc., Boston, MA (1999)

  14. Helgason, S.: Fundamental solutions of invariant differential operators on symmetric spaces. Am. J. Math 86, 565–601 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ionescu, A.D.: An endpoint estimate for the kunze-stein phenomenon and related maximal operators. Ann. Math. (2) 152(1), 259–275 (2000)

    Article  MATH  Google Scholar 

  16. Ishikawa, S.: The range characterizations of the totally geodesic radon transform on the real hyperbolic space. Duke Math. J. 90(1), 149–203 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kumar, A., Ray, S.K.: Weighted estimates of d-plane transform for radial functions on euclidean spaces. Israel J. Math. 188, 25–56 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Markoe, A.: Analytic tomography. Encyclopedia of mathematics and its applications. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  19. Oberlin, D.M., Stein, E.M.: Mapping properties of the radon transform. Indiana Univ. Math. J. 31, 641–650 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  20. Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds, 2nd edn. Graduate Texts in Mathematics, vol. 149. Springer, New York (2006)

  21. Ray, S.K., Sarkar, R.P.: Fourier and radon transform on harmonic \(na\) groups. Trans. Am. Math. Soc. 361(8), 4269–4297 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  22. Rubin, B.: Inversion formulas for the spherical radon transform and the generalized cosine transform. Adv. Appl. Math. 29(3), 471–497 (2002)

    Article  MATH  Google Scholar 

  23. Rubin, B.: Radon, cosine and sine transforms on real hyperbolic space. Adv. Math. 170(2), 206–223 (2002)

    MATH  MathSciNet  Google Scholar 

  24. Rubin, B.: Radon transforms on affine grassmannians. Trans. Am. Math. Soc. 356(12), 5045–5070 (2004)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Strichartz, R.S.: Harmonic analysis on grassmannian bundles. Trans. Am. Math. Soc. 296, 387–409 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  27. Strichartz, R.S.: \(l^p\) estimates for radon transforms in euclidean and non-euclidean spaces. Duke Math. J. 48(4), 699–727 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  28. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Mathematical Series, vol. 32. Princeton University Press, Princeton, NJ (1971)

  29. Vilenkin, N.Ja., Klimyk, A.U.: Representation of Lie groups and special functions, vol. 2. Class I representations, special functions, and integral transforms. Translated from the Russian by V.A. Groza and A. A. Groza. Mathematics and its Applications (Soviet Series), 74. Kluwer Academic Publishers Group, Dordrecht (1993)

Download references

Acknowledgments

Authors would like to thank the referees for several valuable suggestions and criticisms which improved the paper. The authors would like to thank G. Santhanam for illuminating conversations regarding this work.

Author information

Authors and Affiliations

  1. Department of mathematics, Indian Institute of Science, Bangalore , 560012, Karnataka, India

    Ashisha Kumar

  2. Discipline of Mathematics, Indian Institute of Technology Indore, Indore, Madhya Pradesh, India

    Ashisha Kumar

  3. Stat-Math Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata , 700108, India

    Swagato K. Ray

Authors
  1. Ashisha Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Swagato K. Ray
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Swagato K. Ray.

Additional information

Communicated by K. Gröchenig.

The first author is supported by Dr. DS Kothari post doctoral fellowship (No. F.4-2/2006(BSR)/13-396/2010(BSR)).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kumar, A., Ray, S.K. End point estimates for Radon transform of radial functions on non-Euclidean spaces. Monatsh Math 174, 41–75 (2014). https://doi.org/10.1007/s00605-014-0620-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-014-0620-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation