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Mathematische Zeitschrift

, Volume 293, Issue 3–4, pp 1277–1285 | Cite as

Orthogonal testing families and holomorphic extension from the sphere to the ball

  • Luca Baracco
  • Martino FassinaEmail author
Article
  • 106 Downloads

Abstract

Let \(\mathbb {B}^2\) denote the open unit ball in \(\mathbb {C}^2\), and let \(p\in \mathbb {C}^2\)\\(\overline{\mathbb {B}^2}\). We prove that if f is an analytic function on the sphere \(\partial \mathbb {B}^2\) that extends holomorphically in each variable separately and along each complex line through p, then f is the trace of a holomorphic function in the ball.

Keywords

Analytic discs Holomorphic extension Testing families 

Mathematics Subject Classification

Primary 32V25 Secondary 32V20 32V40 

Notes

References

  1. 1.
    Agranovsky, M., Val’sky, R.: Maximality of invariant algebras of functions. Sibirsk. Mat. Z̆. 12, 3–12 (1971)MathSciNetGoogle Scholar
  2. 2.
    Agranovsky, M., Semenov, A.M.: Boundary analogues of the Hartogs theorem. Sibirsk. Mat. Z̆. 12(1), 168–170 (1991). (translation in Siberian Math. J.32 (1991), no. 1)MathSciNetGoogle Scholar
  3. 3.
    Agranovsky, M.: Analog of a theorem of Forelli for boundary values of holomorphic functions on the unit ball of \(\mathbb{C}^n\). J. Anal. Math. 113, 293–304 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Baracco, L., Tumanov, A., Zampieri, G.: Extremal discs and holomorphic extension from convex hypersurfaces. Ark. Mat. 45(1), 1–13 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baracco, L.: Holomorphic extension from the sphere to the ball. J. Math. Anal. Appl. 388(2), 760–762 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Baracco, L.: Separate holomorphic extension along lines and holomorphic extension from the sphere to the ball. Am. J. Math. 135(2), 493–497 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Baracco, L.: Holomorphic extension from a convex hypersurface. Asian J. Math. 20(2), 263–266 (2016)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Baracco, L., Pinton, S.: Testing families of complex lines for the unit ball. J. Math. Anal. Appl. 458(2), 1449–1455 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dinh, T.-C.: Conjecture de Globevnik-Stout et théorème de Morera pour une chaîne holomorphe. Ann. Fac. Sci. Touluse Math. 8(2), 235–257 (1999)CrossRefGoogle Scholar
  10. 10.
    Globevnik, J.: Small families of complex lines for testing holomorphic extendibility. Am. J. Math. 134(6), 1473–1490 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Globevnik, J.: Meromorphic extensions from small families of circles and holomorphic extensions from spheres. Trans. Am. Math. Soc. 364(11), 5857–5880 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hanges, N., Trèves, F.: Propagation of holomorphic extendability of CR functions. Math. Ann. 263(2), 157–177 (1983)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hartogs, F.: Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten. Math. Ann. 62(1), 1–88 (1906)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lawrence, M.G.: Hartog’s separate analyticity theorem for CR functions. Internat. J. Math. 18(3), 219–229 (2007)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lawrence, M.G.: The \(L^p\) CR Hartogs separate analyticity theorem for convex domains. Math. Z. 288(1–2), 401–414 (2018)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. France 109(4), 427–474 (1981)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rudin, W.: Function Theory in the Unit Ball of \(\mathbb{C}^n\), Grundlehren Math. Wiss., vol. 241. Springer, New York (1980)CrossRefGoogle Scholar
  18. 18.
    Stout, E.L.: The boundary values of holomorphic functions of several complex variables. Duke Math J. 44(1), 105–108 (1977)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Tumanov, A: Extremal discs and the geometry of CR manifold. Real methods in complex and CR geometry, pp. 191–212, Lecture Notes in Math, 1848, Springer, Berlin (2004)Google Scholar
  20. 20.
    Tumanov, A.: Testing analyticity on circles. Am. J. Math. 129(3), 785–790 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica Tullio Levi-CivitaUniversità di PadovaPaduaItaly
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

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