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

  • Luca Baracco
  • Martino FassinaEmail author


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.


Analytic discs Holomorphic extension Testing families 

Mathematics Subject Classification

Primary 32V25 Secondary 32V20 32V40 



  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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Baracco, L., Tumanov, A., Zampieri, G.: Extremal discs and holomorphic extension from convex hypersurfaces. Ark. Mat. 45(1), 1–13 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Baracco, L.: Holomorphic extension from the sphere to the ball. J. Math. Anal. Appl. 388(2), 760–762 (2012)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Baracco, L.: Holomorphic extension from a convex hypersurface. Asian J. Math. 20(2), 263–266 (2016)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  10. 10.
    Globevnik, J.: Small families of complex lines for testing holomorphic extendibility. Am. J. Math. 134(6), 1473–1490 (2012)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hanges, N., Trèves, F.: Propagation of holomorphic extendability of CR functions. Math. Ann. 263(2), 157–177 (1983)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lawrence, M.G.: Hartog’s separate analyticity theorem for CR functions. Internat. J. Math. 18(3), 219–229 (2007)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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

Personalised recommendations