Advertisement

Complex Analysis and Operator Theory

, Volume 6, Issue 2, pp 491–513 | Cite as

The Szegö Metric Associated to Hardy Spaces of Clifford Algebra Valued Functions and Some Geometric Properties

  • Dennis Grob
  • Rolf Sören KraußharEmail author
Article
  • 85 Downloads

Abstract

In analogy to complex function theory we introduce a Szegö metric in the context of hypercomplex function theory dealing with functions that take values in a Clifford algebra. In particular, we are dealing with Clifford algebra valued functions that are annihilated by the Euclidean Dirac operator in \({\mathbb{R}^{m+1}}\) . These are often called monogenic functions. As a consequence of the isometry between two Hardy spaces of monogenic functions on domains that are related to each other by a conformal map, the generalized Szegö metric turns out to have a pseudo-invariance under Möbius transformations. This property is crucially applied to show that the curvature of this metric is always negative on bounded domains. Furthermore, it allows us to establish that this metric is complete on bounded domains.

Keywords

Bounded Domain Dirac Operator Hardy Space Bergman Space Clifford Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahlfors L.: Clifford numbers and Möbius transformations in \({\mathbb{R}^n}\) . In: Chisholm, J.S.R., Common, A.K. (eds) Clifford Algebras and Their Applications in Mathematical Physics. NATO ASI Series, Series C: Mathematical and Physical Sciences, vol. 183, pp. 167–175. D. Reidel, Dordrecht-Boston-Lancaster-Tokyo (1986)Google Scholar
  2. 2.
    Bergman, S.: The kernel function and conformal mapping. Mathematical Surveys, V, VII. American Mathematical Society (AMS), New York (1950)Google Scholar
  3. 3.
    Bergman, S.: Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande. J. Reine Angew. Math. 169, 1–42 (1933); 172, 89–128 (1935)Google Scholar
  4. 4.
    Błocki Z., Pflug P.: Hyperconvexity and Bergman completeness. Nagoya Math. J. 151, 221–225 (1998)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Blumberg, S.: Das Randverhalten des Bergman-Kerns und der Bergman-Metrik auf lineal konvexen Gebieten endlichen Typs, Ph.D. thesis, Fachbereich C-Mathematik und Naturwissenschaften, Bergische Universität Wuppertal (2005)Google Scholar
  6. 6.
    Brackx F., Delanghe R.: Hypercomplex function theory and Hilbert modules with reproducing kernel. Proc. Lond. Math. Soc. 37, 545–576 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis. Pitman Research Notes, vol. 76. Boston (1982)Google Scholar
  8. 8.
    Cervantes J., Gonzáles, O.: Sobre Los Espacios De Bergman En Bajas Dimensiones. Ph.D. Thesis, Instituto Politécnico Nacional (2009)Google Scholar
  9. 9.
    Chen B.-Y.: Bergman completeness of hyperconvex manifolds. Nagoya Math. J. 175, 165–170 (2004)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Chen B.-Y.: The Bergman metric on complete Kähler manifolds. Math. Ann. 327(2), 339–349 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Chen B.-Y.: Completeness of the Bergman metric on non-smooth pseudoconvex domains. Ann. Pol. Math. 71(3), 241–251 (1999)zbMATHGoogle Scholar
  12. 12.
    Cnops, J.: Hurwitz pairs and applications of Möbius transformations. Ph.D. Thesis, Ghent State University (1993/1994)Google Scholar
  13. 13.
    Cnops, J.: Introduction to Dirac operators on manifolds. Progress in Mathematical Physics, vol. 24. Birkhäuser, Boston (2002)Google Scholar
  14. 14.
    Constales, D.: The relative position of L 2 domains in complex and Clifford analysis. Ph.D. Thesis, Ghent State University (1989/1990)Google Scholar
  15. 15.
    Delanghe R., Sommen F., Souček V.: Clifford Algebra and Spinor Valued Functions. Kluwer, Dordrecht (1992)zbMATHCrossRefGoogle Scholar
  16. 16.
    Diederich K.: Das Randverhalten der Bergmanschen Kernfunktion und Metrik in streng pseudokonvexen Gebieten. Math. Ann. 187, 9–36 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Diederich, K., Fornæss, F., Herbort, G.: Boundary behavior of the Bergman metric. Complex Analysis of Several Variables. Proc. Symp., Madison/Wis. 1982. Proc. Symp. Pure Math., vol. 41, pp. 59–67 (1984)Google Scholar
  18. 18.
    Fischer, W., Lieb, I.: Aufgewählte Kapitel der Funktionentheorie. Vieweg Verlag (1988)Google Scholar
  19. 19.
    Fornæss J.E., Lee L.: Kobayashi, Carathodory and Sibony metric. Complex Var. Elliptic Equ. 54(3–4), 293–301 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gürlebeck K., Habetha K., Sprößig W.: Holomorphic Functions in the Plane and n-Dimensional Space. Birkhäuser, Boston (2008)zbMATHGoogle Scholar
  21. 21.
    Gürlebeck K., Sprößig W.: Quaternionic Analysis and Elliptic Boundary Value Problems. Birkhäuser, Boston (1990)zbMATHCrossRefGoogle Scholar
  22. 22.
    Hahn K.T., Kyong T.: On completeness of the Bergman metric and its subordinate metric. Proc. Natl. Acad. Sci. USA 73, 4294 (1976)zbMATHCrossRefGoogle Scholar
  23. 23.
    Hahn K.T., Kyong T.: On completeness of the Bergman metric and its subordinate metric, II. Pac. J. Math. 68, 437–446 (1977)zbMATHGoogle Scholar
  24. 24.
    Hahn K.T., Kyong T.: Inequality between the Bergman metric and the Carathéodory differential metric. Proc. Am. Math. Soc. 68, 193–194 (1978)zbMATHGoogle Scholar
  25. 25.
    Herbort G.: On the problem of Kähler convexity in the Bergman metric. Mich. Math. J. 52(3), 543–552 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Herbort G.: The Bergman metric on hyperconvex domains. Math. Z. 232, 183–196 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Heuser, H.: Funktionalanalysis—Theorie und Anwendung, 4th edn. Vieweg+Teubner Verlag (2006)Google Scholar
  28. 28.
    Jarnicki, M., Pflug, P., Zwonek, W.: On Bergman completeness of non-hyperconvex domains. Zesz. Nauk. Uniw. Jagiell. 1245. Univ. Iagell. Acta Math. 38, 169–184 (2000)Google Scholar
  29. 29.
    Kim, K.T., Krantz, S.: The Bergman metric invariants and their boundary behavior. In: Bland, J., et al. (eds.) Explorations in Complex and Riemannian Geometry. A Volume dedicated to Robert E. Greene. Contemp. Math., vol. 332, pp. 139–151. American Mathematical Society (AMS), Providence (2003)Google Scholar
  30. 30.
    Kobayashi S.: Geometry of bounded domains. Trans. Am. Math. Soc. 92, 267–290 (1959)zbMATHCrossRefGoogle Scholar
  31. 31.
    Kobayashi S.: On complete Bergman metrics. Proc. Am. Math. Soc. 13(4), 511–513 (1962)zbMATHCrossRefGoogle Scholar
  32. 32.
    Krantz, S.: Complex Analysis: The Geometric Viewpoint, 2nd edn. The Mathematical Association of America (2004)Google Scholar
  33. 33.
    Krantz, S.: Function Theory of Several Complex Variables, 2nd edn. AMS Chealsea Publishing (2001)Google Scholar
  34. 34.
    Krantz S.: The Carathéodory and Kobayashi metrics and applications in complex analysis. Am. Math. Mon. 115(4), 304–329 (2008)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Kraußhar, R.S.: Conformal mappings and Szegö kernels in Quaternions. Diploma thesis, Lehrstuhl II für Mathematik, RWTH Aachen (1998)Google Scholar
  36. 36.
    Lieder, M.: Das Randverhalten der Kobayashi- und Carathéodory-Metrik auf lineal konvexen Gebieten endlichen Typs. Ph.D. thesis, Bergische Universität Wuppertal (2005)Google Scholar
  37. 37.
    Nikolov N.: The completeness of the Bergman distance of planar domains has a local character. Complex Variables Theory Appl. 48(8), 705–709 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Nikolov N., Pflug P.: Estimates for the Bergman kernel and metric of convex domains in \({{\mathbb C}^n}\) . Ann. Pol. Math. 81(1), 73–78 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Nikolov N., Pflug P.: Behavior of the Bergman kernel and metric near convex boundary points. Proc. Am. Math. Soc. 131(7), 2097–2102 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Ryan J.: Intertwining operators for iterated Dirac operators over Minkowski-type spaces. J. Math. Anal. Appl. 177, 1–23 (1993)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Lehrstuhl A für MathematikRheinisch-Westfälische Technische Hochschule AachenAachenGermany
  2. 2.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany

Personalised recommendations