Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 401–414 | Cite as

The \(L^p\) CR Hartogs separate analyticity theorem for convex domains

  • Mark G. Lawrence


In this note, a very general theorem of the CR Hartogs type is proved for almost generic strictly convex domains in \(\mathbf{C}^{\mathbf{n}}\) with real analytic boundary. Given such a domain D, and given an \(L^p\) function f on \(\partial D\) which has holomorphic extensions on the slices of D by complex lines parallel to the coordinate axes, f must be CR—i.e. f has a holomorphic extension to D which is in the Hardy space \(H^p(D)\). This is the first general result of CR Hartogs type which is not for the ball, or some other domain with symmetries, and holds for \(L^1\) functions. As a corollary, the Szegő kernel is shown to be the strong operator limit of \((\pi _1 \pi _2)^n\), where \(\pi _i\) is the projection onto \(z_i\) holomorphically extendible \(L^2(\partial D)\) functions (in \(\mathbf{C}^{\mathbf{2}}\), with a slightly more complicated formula in \(\mathbf{C}^{\mathbf{n}}\)).


  1. 1.
    Agranovsky, M.L.: Propagation of boundary CR foliations and Morera type theorems for manifolds with attached analytic discs. Adv. Math. 211(1), 284–326 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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
  3. 3.
    Baracco, L.: Holomorphic extension from a convex hypersurface. Asian J. Math. 20(2), 263–266 (2016)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.
    Bertrand, F., Gong, X.: Dirichlet and Neumann problems for planar domains with parameter. Trans. Am. Math. Soc. 366(1), 159–217 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Boggess, A.: CR Manifolds and the Tangential Cauchy–Riemann Complex. Studies in Advanced Mathematics. CRC Press, Boca Raton (1991)zbMATHGoogle Scholar
  7. 7.
    Dinh, T.-C.: Conjecture de Globevnik-Stout et théorème de Morera pour une chaîne holomorphe. Ann. Fac. Sci. Toulouse Math. (6) 8(2), 235–257 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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
  9. 9.
    Kytmanov, A. M., Aĭzenberg, L. A.: The holomorphy of continuous functions that are representable by the Martinelli–Bochner integral. Izv. Akad. Nauk Armjan. SSR Ser. Mat. 13(2), 158–169, 173 (1978)Google Scholar
  10. 10.
    Kytmanov, A. M.: The Bochner–Martinelli Integral and Its Applications. Birkhäuser Verlag, Basel (1995). Translated from the Russian by Harold P. Boas and revised by the authorGoogle Scholar
  11. 11.
    Kytmanov, A.M., Myslivets, S.G.: An analog of the Hartogs theorem in a ball of \({{\bf C}}^{{\bf n}}\). Math. Nachr. 288(2–3), 224–234 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lawrence, M.G.: Hartogs’ separate analyticity theorem for CR functions. Int. J. Math. 18(3), 219–229 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Stout, E.L.: The boundary values of holomorphic functions of several complex variables. Duke Math. J. 44(1), 105–108 (1977)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.AstanaKazakhstan

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