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Riesz Means on Locally Symmetric Spaces

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Abstract

We prove that for a certain class of n dimensional rank one locally symmetric spaces, if \(f \in L^p\), \(1\le p \le 2\), then the Riesz means of order z of f converge to f almost everywhere, for \(\mathrm {Re}z> (n-1)(1/p-1/2)\).

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References

  1. Alexopoulos, G., Lohoué, N.: Riesz means on Lie groups and Riemannian manifolds of nonnegative curvature. Bull. Soc. Math. France 122(2), 209–223 (1994)

    Article  MathSciNet  Google Scholar 

  2. Anker, J.-Ph., Damek, E., Yacoub, Ch.: Spherical analysis on harmonic \(NA\) groups. Annali Scuola Norm. Sup. di Pisa 23(4), 643–679 (1996)

    MathSciNet  MATH  Google Scholar 

  3. Anker, J.-Ph., Ji, L.: Heat kernel and Green function estimates on noncompact symmetric spaces. Geom. Funct. Anal. 9(6), 1035–1091 (1999)

    Article  MathSciNet  Google Scholar 

  4. Anker, J.-Ph., Ostellari, P.: The heat kernel on noncompact symmetric spaces. Am. Math. Soc. Transl. Ser. 2 210, 27–46 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Berard, P.: Riesz means on Riemannian manifolds. Proc. Sympos. Pure Math. 36, 1–12 (1980)

    Article  MathSciNet  Google Scholar 

  6. Christ, M.: Weak type endpoint bounds for Bochner–Riesz operators. Rev. Mat. Iberoam. 3, 25–31 (1987)

    Article  Google Scholar 

  7. Christ, M.: Weak type \((1,1)\) bounds for rough operators. Ann. Math. 128(2), 19–42 (1988)

    Article  MathSciNet  Google Scholar 

  8. Christ, M., Sogge, C.: Weak type \(L^1\) convergence of eigenfunction expansions for pseudodifferential operators. Invent. Math. 94, 421–453 (1988)

    Article  MathSciNet  Google Scholar 

  9. Clerc, J.L., Stein, E.M.: \(L^{p}\) multipliers for noncompact symmetric spaces. Proc. Nat. Acad. Sci. U. S. A. 71, 3911–3912 (1974)

    Article  MathSciNet  Google Scholar 

  10. Corlette, K.: Hausdorff dimensions of limit sets. I. Invent. Math. 102(3), 521–541 (1990)

    Article  MathSciNet  Google Scholar 

  11. Cowling, M.G.: Harmonic analysis on semigroups. Ann. Math. 117, 267–283 (1983)

    Article  MathSciNet  Google Scholar 

  12. Davies, E.B., Mandouvalos, N.: Heat kernel bounds on hyperbolic space and Kleinian groups. Proc. London Math. Soc. (3) 57(1), 182–208 (1988)

    Article  MathSciNet  Google Scholar 

  13. Davis, K., Chang, Y.: Lectures on Bochner–Riesz Means (London Mathematical Society Lecture Note Series). Cambridge University Press, Cambridge (1987)

    Book  Google Scholar 

  14. Fotiadis, A., Papageorgiou, E.: Riesz means on symmetric spaces. J. Math. Anal. Appl. 499(1), 124970 (2021)

    Article  MathSciNet  Google Scholar 

  15. Giulini, S., Mauceri, G.: Almost everywhere convergence of Riesz means on certain noncompact symmetric spaces. Ann. di Mat. Pura ed Appl. 159, 357–369 (1991)

    Article  MathSciNet  Google Scholar 

  16. Giulini, S., Travaglini, G.: Estimates for Riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds. J. Funct. Anal. 96, 1–30 (1991)

    Article  MathSciNet  Google Scholar 

  17. Grafakos, L.: Classical and Modern Fourier Analysis. Pearson Education, New Jersey (2004)

    MATH  Google Scholar 

  18. Grigor’yan, A.: Heat Kernel and Analysis on Manifolds, vol. 47. American Mathematical Society (2009)

  19. Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, 1st edn. Academic Press, New York (1978)

    MATH  Google Scholar 

  20. Helgason, S.: Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs, 2nd edn. American Mathematical Society (2008)

  21. Hörmander, L.: On the Riesz Means of Spectral Functions and Eigenfunction Expansions for Elliptic Differential Operators. Some Recent Advances in the Basic Sciences, pp. 155–202. Yeshiva University, New York (1966)

    Google Scholar 

  22. Knieper, G.: On the asymptotic geometry of nonpositively curved manifolds. Geom. Funct. Anal. 7, 755–782 (1997)

    Article  MathSciNet  Google Scholar 

  23. Lizhen, J., Li, P., Wang, J.: Ends of locally symmetric spaces with maximal bottom spectrum. J. Reine Angew. Math. 632, 1–35 (2009)

    MathSciNet  MATH  Google Scholar 

  24. Leuzinger, E.: Critical exponents of discrete groups and \(L^2\)-spectrum. Proc. Am. Math. Soc. 132(3), 919–927 (2004)

    Article  MathSciNet  Google Scholar 

  25. Nicholls, P.: The Ergodic Theory of Discrete Groups (London Mathematical Society Lecture Note Series). Cambridge University Press, Cambridge (1989)

    Book  Google Scholar 

  26. Sogge, C.: On the convergence of Riesz means on compact manifolds. Ann. Math. 126(2), 439–447 (1987)

    Article  MathSciNet  Google Scholar 

  27. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  28. Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, (AM-63), vol. 63. Princeton University Press, Princeton (1971)

    Google Scholar 

  29. Yue, C.: The ergodic theory of discrete isometry groups on manifolds of variable negative curvature Trans. Am. Math. Soc. 348, 4965–5005 (1996)

    Article  Google Scholar 

  30. Zhang, H.-W.: Wave and Klein–Gordon equations on certain locally symmetric spaces. J. Geom. Anal. 30(4), 4386–4406 (2020)

    Article  MathSciNet  Google Scholar 

  31. Zhu, F.: Almost everywhere convergence of Riesz means on noncompact symmetric space \(SL(3, \mathbb{H})/Sp(3)\). Acta Math. Sin. New Ser. 13(4), 545–552 (1997)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The author would like to thank the referee for his/her critical comments and the careful and insightful review, as well as M. Kolountzakis and M. Papadimitrakis for conversations and remarks.

Funding

Supported by the Hellenic Foundation for Research and Innovation, Project HFRI-FM17-1733.

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Author notes

  1. Effie Papageorgiou

    Present address: Department of Mathematics and Applied Mathematics, University of Crete, Voutes Campus, 70013, Heraklion, Crete, Greece

Authors and Affiliations

  1. Department of Mathematics, Aristotle University of Thessaloniki, 54124, Thessaloníki, Greece

    Effie Papageorgiou

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  1. Effie Papageorgiou
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Communicated by Andreas Seeger.

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Papageorgiou, E. Riesz Means on Locally Symmetric Spaces. Complex Anal. Oper. Theory 16, 43 (2022). https://doi.org/10.1007/s11785-022-01226-7

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