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Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres

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Abstract

A sharp \(L^p\) spectral multiplier theorem of Mihlin–Hörmander type is proved for a distinguished sub-Laplacian on quaternionic spheres. This is the first such result on compact sub-Riemannian manifolds where the horizontal space has corank greater than one. The proof hinges on the analysis of the quaternionic spherical harmonic decomposition, of which we present an elementary derivation.

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References

  1. Agrachev, A., Boscain, U., Gauthier, J.-P., Rossi, F.: The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups. J. Funct. Anal. 256(8), 2621–2655 (2009)

    Article  MathSciNet  Google Scholar 

  2. Ahrens, J.: Spectral Decomposition of a Distinct Sub-Laplacian on the Quaternionic Sphere. Christian-Albrechts-Universität zu Kiel, Masterarbeit (2016)

    Google Scholar 

  3. Astengo, F., Cowling, M., Di Blasio, B.: The Cayley transform and uniformly bounded representations. J. Funct. Anal. 213(2), 241–269 (2004)

    Article  MathSciNet  Google Scholar 

  4. Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, second ed., Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York (2001)

    Book  Google Scholar 

  5. Baudoin, F., Wang, J.: The subelliptic heat kernels of the quaternionic Hopf fibration. Potential Anal. 41(3), 959–982 (2014)

    Article  MathSciNet  Google Scholar 

  6. Bellaïche, A.: The tangent space in sub-Riemannian geometry, In: Bellaïche, A., Risler, J.-J. (eds.) Sub-Riemannian Geometry, Progr. Math., vol. 144, Birkhäuser, Basel, pp. 1–78 (1996)

    Chapter  Google Scholar 

  7. Biquard, O.: Quaternionic contact structures, In: Marchiafava, S., Piccinni, P., Pontecorvo, M. (eds) Quaternionic structures in mathematics and physics (Rome, 1999), Univ. Studi Roma “La Sapienza”, Rome, pp. 23–30 (1999)

  8. Casarino, V., Ciatti, P.: \(L^p\) joint eigenfunction bounds on quaternionic spheres. J. Fourier Anal. Appl. 23(4), 886–918 (2017)

    Article  MathSciNet  Google Scholar 

  9. Casarino, V., Cowling, M.G., Martini, A., Sikora, A.: Spectral multipliers for the Kohn Laplacian on forms on the sphere in \({\mathbb{C}}^n\). J. Geom. Anal. 27(4), 3302–3338 (2017)

    Article  MathSciNet  Google Scholar 

  10. Cowling, M.G., Martini, A.: Sub-Finsler geometry and finite propagation speed, In: Picardello, M.A. (ed.) Trends in Harmonic Analysis, Springer INdAM Series, vol. 3, pp. 147–205. Springer, Milan (2013)

    Chapter  Google Scholar 

  11. Cowling, M.G., Sikora, A.: A spectral multiplier theorem for a sublaplacian on \(\text{ SU(2) }\). Math. Z. 238(1), 1–36 (2001)

    Article  MathSciNet  Google Scholar 

  12. Cowling, M.G., Klima, O., Sikora, A.: Spectral multipliers for the Kohn sublaplacian on the sphere in \(\mathbb{C}^n\). Trans. Amer. Math. Soc. 363(2), 611–631 (2011)

    Article  MathSciNet  Google Scholar 

  13. Duong, X.T., Ouhabaz, E.M., Sikora, A.: Plancherel-type estimates and sharp spectral multipliers. J. Funct. Anal. 196(2), 443–485 (2002)

    Article  MathSciNet  Google Scholar 

  14. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. I. Robert E. Krieger Publishing Co. Inc., Melbourne, Fla (1981)

  15. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. II. Robert E. Krieger Publishing Co. Inc., Melbourne, Fla (1981)

  16. Fefferman, C., Phong, D.H.: Subelliptic eigenvalue problems. In: Beckner, W., Calderón, A.P., Fefferman, R., Jones P.W. (eds.) Conference on Harmonic Analysis in Honor of Antoni Zygmund, vol. II (Chicago, Ill., 1981). Wadsworth Math. Ser., Wadsworth, Belmont, CA, pp. 590–606 (1983)

  17. Folland, G.B.: The tangential Cauchy-Riemann complex on spheres. Trans. Amer. Math. Soc. 171, 83–133 (1972)

    Article  MathSciNet  Google Scholar 

  18. Hall, B.C.: Lie Groups, Lie Algebras, and Representations, Graduate Texts in Mathematics, vol. 222. Springer-Verlag, New York (2003)

    Book  Google Scholar 

  19. Hebisch, W.: Functional calculus for slowly decaying kernels, preprint (1995)

  20. Hebisch, W.: Multiplier theorem on generalized Heisenberg groups. Colloq. Math. 65(2), 231–239 (1993)

    Article  MathSciNet  Google Scholar 

  21. Johnson, K.D., Wallach, N.R.: Composition series and intertwining operators for the spherical principal series. I. Trans. Amer. Math. Soc. 229, 137–173 (1977)

    Article  MathSciNet  Google Scholar 

  22. Martini, A., Müller, D., Nicolussi Golo, S.: Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type, preprint (2018). arXiv:1812.02671

  23. Martini, A.: Spectral multipliers on Heisenberg-Reiter and related groups. Ann. Mat. Pura Appl. 194(4), 1135–1155 (2015)

    Article  MathSciNet  Google Scholar 

  24. Martini, A.: Joint functional calculi and a sharp multiplier theorem for the Kohn Laplacian on spheres. Math. Z. 286, 1539–1574 (2017)

    Article  MathSciNet  Google Scholar 

  25. Martini, A., Müller, D.: Spectral multiplier theorems of Euclidean type on new classes of \(2\)-step stratified groups. Proc. Lond. Math. Soc. (3) 109(5), 1229–1263 (2014)

    Article  MathSciNet  Google Scholar 

  26. Martini, A., Müller, D.: Spectral multipliers on \(2\)-step groups: topological versus homogeneous dimension. Geom. Funct. Anal. 26(2), 680–702 (2016)

    Article  MathSciNet  Google Scholar 

  27. Melrose, R.: Propagation for the wave group of a positive subelliptic second-order differential operator. In: Mizohata, S. (ed.) Hyperbolic Equations and Related Topics (Katata/Kyoto, 1984), pp. 181–192. Academic Press, Boston (1986)

    Chapter  Google Scholar 

  28. Montgomery, R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, vol. 91. American Mathematical Society, Providence, RI (2002)

    Google Scholar 

  29. Müller, D., Stein, E.M.: On spectral multipliers for Heisenberg and related groups. J. Math. Pures Appl. (9) 73(4), 413–440 (1994)

    MathSciNet  MATH  Google Scholar 

  30. Nagel, A., Rudin, W.: Moebius-invariant function spaces on balls and spheres. Duke Math. J. 43(4), 841–865 (1976)

    Article  MathSciNet  Google Scholar 

  31. Pajas, P., Raçzka, R.: Degenerate representations of the symplectic groups. I. The compact group. J. Math. Phys. 9, 1188–1201 (1968)

    Article  MathSciNet  Google Scholar 

  32. Seeger, A., Sogge, C.D.: On the boundedness of functions of (pseudo-) differential operators on compact manifolds. Duke Math. J. 59(3), 709–736 (1989)

    Article  MathSciNet  Google Scholar 

  33. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, vol. 32. Princeton University Press, Princeton (1971)

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Authors and Affiliations

  1. Deutsches Forschungszentrum für Künstliche Intelligenz, Trippstadter Str. 122, 67663, Kaiserslautern, Germany

    Julian Ahrens

  2. School of Mathematics, University of New South Wales, UNSW, Sydney, NSW, 2052, Australia

    Michael G. Cowling

  3. School of Mathematics, University of Birmingham, Birmingham, Edgbaston, B15 2TT, UK

    Alessio Martini

  4. Mathematisches Seminar, C.-A.-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24118, Kiel, Germany

    Detlef Müller

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  1. Julian Ahrens
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  2. Michael G. Cowling
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  3. Alessio Martini
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  4. Detlef Müller
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Correspondence to Alessio Martini.

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Cowling was supported by the Australian Research Council, through grant DP140100531. Martini is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). Müller was supported by the Deutsche Forschungsgemeinschaft, through grant MU 761/11-1.

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Ahrens, J., Cowling, M.G., Martini, A. et al. Quaternionic spherical harmonics and a sharp multiplier theorem on quaternionic spheres. Math. Z. 294, 1659–1686 (2020). https://doi.org/10.1007/s00209-019-02313-w

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  • DOI: https://doi.org/10.1007/s00209-019-02313-w

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