Skip to main content
SpringerLink
Log in

The Poisson transform on a compact real analytic Riemannian manifold

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

We study the transform mapping an \(L^{2}\) function f on a compact, real analytic Riemannian manifold X to the analytic continuation of \(\exp (-t \sqrt{\Delta })f\) to the interior of a Grauert tube tube \(M_{t}\) about X. We show that after precomposing with an elliptic pseudodifferential operator this becomes a unitary map from \(L^{2}(X)\) onto the holomorphic \(L^{2}\) functions on \(M_{t}\). If a compact Lie group of isometries acts transitively on X then the inverse of this unitarized map can be constructed by the restrict ion principle.

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

Notes

  1. Note that \(\mathcal {O}^{\mathrm {Id}}(\partial M_{\epsilon })\) is not a reproducing kernel Hilbert space because for fixed \(z_{0}\in \partial \Omega \), the Szëgo kernel \(S(\cdot ,z_{0})\) is not in \(L^{2}(\partial \Omega )\). Reproducing kernel Hilbert spaces are useful for many applications including phase space bounds, coherent states, and Berezin quantization.

  2. A bounded positive operator is automatically self-adjoint. We will require unbounded positive operators to be self-adjoint as part of the definition of positivity.

  3. We recall there is an \(\epsilon _{0}>0\) such that for all \(\epsilon \in (0, \epsilon _{0})\), all the eigenfunctions of the Laplacian can be analytically continued to \(M_{\epsilon }\).

  4. The map is bounded because \(J_{\epsilon }(k)^{-1/2} \sim c_{n, \epsilon } e^{-\epsilon |k|}(|k|^{(n+1)/4}+\text {lower order terms})\) (using stationary phase for complex valued phase functions; see [23, Theorem 2.3]).

References

  1. Akhiezer, D.N., Gindikin, S.G.: On Stein extensions of real symmetric spaces. Math. Ann. 286(1–3), 1–12 (1990). doi:10.1007/BF01453562

    Article  MATH  MathSciNet  Google Scholar 

  2. Bérard-Bergery, L., Bourguignon, J.-P.: Laplacians and Riemannian submersions with totally geodesic fibres. Ill. J. Math. 26(2), 181–200 (1982)

    MATH  Google Scholar 

  3. Boutet de Monvel, L: Convergence dans le domaine complexe des séries de fonctions propres. C. R. Acad. Sci. Paris Sér. A-B 287(13), A855–A856 (1978). (French, with English summary)

  4. Boutet de Monvel, L.: On the index of Toeplitz operators of several complex variables. Invent. Math. 50(3), 249–272 (1978/79)

  5. Davidson, M., Ólafsson, G.: The generalized Segal-Bargmann transform and special functions. Acta Appl. Math. 81(1–3), 29–50 (2004). doi:10.1023/B:ACAP.0000024193.17395.d7

    Article  MATH  MathSciNet  Google Scholar 

  6. Davidson, M., Ólafsson, G., Zhang, G.: Laplace and Segal–Bargmann transforms on Hermitian symmetric spaces and orthogonal polynomials. J. Funct. Anal. 204(1), 157–195 (2003). doi:10.1016/S0022-1236(03)00101-0

    Article  MATH  MathSciNet  Google Scholar 

  7. Davidson, M., Ólafsson, G., Zhang, G.: Laguerre polynomials, restriction principle, and holomorphic representations of \({\rm SL}(2,{\mathbf{R}})\). Acta Appl. Math. 71(3), 261–277 (2002). doi:10.1023/A:1015283100541

    Article  MATH  MathSciNet  Google Scholar 

  8. Griffel, D.H.: Applied Functional Analysis, 1985th edn. Dover Publications Inc., Mineola (2002)

    MATH  Google Scholar 

  9. Guillemin, V., Stenzel, M.: Grauert tubes and the homogeneous Monge–Ampère equation. J. Differ. Geom. 34(2), 561–570 (1991)

    MATH  MathSciNet  Google Scholar 

  10. Hall, B.C.: The Segal–Bargmann “coherent state” transform for compact Lie groups. J. Funct. Anal. 122(1), 103–151 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  11. Hall, B.C.: The inverse Segal–Bargmann transform for compact Lie groups. J. Funct. Anal. 143(1), 98–116 (1997). doi:10.1006/jfan.1996.2954

    Article  MATH  MathSciNet  Google Scholar 

  12. Hall, B.C.: Phase space bounds for quantum mechanics on a compact Lie group. Commun. Math. Phys. 184(1), 233–250 (1997). doi:10.1007/s002200050059

    Article  MATH  Google Scholar 

  13. Hall, B.C., Mitchell, J.J.: The Segal–Bargmann transform for noncompact symmetric spaces of the complex type. J. Funct. Anal. 227(2), 338–371 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hall, B.C., Mitchell, J.J.: Isometry theorem for the Segal–Bargmann transform on a noncompact symmetric space of the complex type. J. Funct. Anal. 254(6), 1575–1600 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hall, B.C., Mitchell, J.J.: The Segal–Bargmann transform for compact quotients of symmetric spaces of the complex type. Taiwan. J. Math. 16(1), 13–45 (2012)

    MATH  MathSciNet  Google Scholar 

  16. Hilgert, J., Zhang, G.: Segal–Bargmann and Weyl transforms on compact Lie groups. Monatsh. Math. 158(3), 285–305 (2009). doi:10.1007/s00605-008-0080-0

    Article  MATH  MathSciNet  Google Scholar 

  17. Hörmander, L.: An introduction to complex analysis in several variables, 3rd ed. In: North-Holland Mathematical Library, vol. 7. North-Holland Publishing Co., Amsterdam (1990)

  18. Krötz, B., Ólafsson, G., Stanton, R.J.: The image of the heat kernel transform on Riemannian symmetric spaces of the noncompact type. Int. Math. Res. Not. 22, 1307–1329 (2005)

    Article  Google Scholar 

  19. Krötz, B., Thangavelu, S., Xu, Y.: Heat kernel transform for nilmanifolds associated to the Heisenberg group. Rev. Mat. Iberoam. 24(1), 243–266 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lebeau, G.: FBI transform and the complex Poisson kernel on a compact analytic Riemannian manifold. In: Minicourse at the Workshop on Microlocal Analysis, Northwestern University (2013)

  21. Lempert, L., Szöke, R.: Global solutions of the homogeneous complex Monge–Ampère equation and complex structures on the tangent bundle of Riemannian manifolds. Math. Ann. 290(4), 689–712 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  22. Luo, S.-L.: On the Bargmann transform and the Wigner transform. Bull. Lond. Math. Soc. 30(4), 413–418 (1998). doi:10.1112/S0024609398004457

    Article  MATH  Google Scholar 

  23. Melin, A., Sjöstrand, J.: Fourier integral operators with complex-valued phase functions. In: Fourier Integral Operators and Partial Differential Equations (Colloq. Internat., Univ. Nice, Nice, 1974), Lecture Notes in Math., vol. 459, pp. 120–223. Springer, Berlin (1975)

  24. Ólafsson, G., Ørsted, B.: Generalizations of the Bargmann transform. In: Lie Theory and its Applications in Physics (Clausthal, 1995), pp. 3–14. World Sci. Publ., River Edge (1996)

  25. Seeley, R.T.: Complex powers of an elliptic operator. In: Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), pp. 288–307. Amer. Math. Soc., Providence (1967)

  26. Sontz, S.B.: The C-version Segal–Bargmann transform for finite Coxeter groups defined by the restriction principle. Adv. Math. Phys., Art. ID 365085, 23 (2011)

  27. Stenzel, M.B.: The Segal–Bargmann transform on a symmetric space of compact type. J. Funct. Anal. 165(1), 44–58 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  28. Stenzel, M.B.: An inversion formula for the Segal–Bargmann transform on a symmetric space of non-compact type. J. Funct. Anal. 240(2), 592–608 (2006). doi:10.1016/j.jfa.2006.06.007

    Article  MATH  MathSciNet  Google Scholar 

  29. Stenzel, M.B.: On the analytic continuation of the Poisson kernel. Manuscr. Math. 144(1–2), 253–276 (2014). doi:10.1007/s00229-013-0653-7

    Article  MATH  MathSciNet  Google Scholar 

  30. Zelditch, S.: Complex zeros of real ergodic eigenfunctions. Invent. Math. 167(2), 419–443 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Zelditch, S.: Index and dynamics of quantized contact transformations. Ann. Inst. Fourier (Grenoble) 47(1), 305–363 (1997). (English, with English and French summaries)

    Article  MATH  MathSciNet  Google Scholar 

  32. Zelditch, S.: Pluri-potential theory on Grauert tubes of real analytic Riemannian manifolds, I. In: Spectral Geometry, Proc. Sympos. Pure Math., vol. 84, pp. 299–339. Amer. Math. Soc., Providence (2012). doi:10.1090/pspum/084/1363

Download references

Author information

Authors and Affiliations

  1. The Ohio State University at Newark, Newark, USA

    Matthew B. Stenzel

Authors
  1. Matthew B. Stenzel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew B. Stenzel.

Additional information

Communicated by A. Cap.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stenzel, M.B. The Poisson transform on a compact real analytic Riemannian manifold. Monatsh Math 178, 299–309 (2015). https://doi.org/10.1007/s00605-015-0798-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-015-0798-4

Keywords

Mathematics Subject Classification

Search

Navigation