Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Extension of CR-functions into weighted wedges through families of nonsmooth analytic discs

Authors: Dmitri Zaitsev and Giuseppe Zampieri
Journal: Trans. Amer. Math. Soc. 356 (2004), 1443-1462
MSC (2000): Primary 32V10, 32V25, 32D15
Published electronically: September 22, 2003
MathSciNet review: 2034313
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The goal of this paper is to develop a theory of nonsmooth analytic discs attached to domains with Lipschitz boundary in real submanifolds of $\mathbb{C} ^{n}$. We then apply this technique to establish a propagation principle for wedge extendibility of CR-functions on these domains along CR-curves and along boundaries of attached analytic discs. The technique from this paper has been also extensively used by the authors recently to obtain sharp results on wedge extension of CR-functions on wedges in prescribed directions extending results of BOGGESS-POLKING and EASTWOOD-GRAHAM.

References [Enhancements On Off] (What's this?)

  • [AH81] Airapetyan, R.A.; Henkin, G.M. -- Analytic continuation of CR-functions through the ``edge of the wedge". Sov. Math., Dokl. 24 (1981), 129-132; translation from Dokl. Akad. Nauk SSSR 259 (1981), 777-781. MR 82k:32039
  • [BER99] Baouendi, M.S.; Ebenfelt, P.; Rothschild, L.P. -- Real Submanifolds in Complex Space and Their Mappings. Princeton Math. Series 47, Princeton Univ. Press, 1999. MR 2000b:32066
  • [BRT94] Baouendi, M.S.; Rothschild, L.P.; Trépreau, J.-M. -- On the geometry of analytic discs attached to real manifolds. J. Differential Geom. 39 (1994), 379-405. MR 95a:32026
  • [BT81] Baouendi, M.S.; Treves, F. -- A property of the functions and distributions annihilated by a locally integrable system of complex vector fields. Ann. Math. (2) 114, 387-421, (1981). MR 82i:35057
  • [BZ01] Baracco, L; Zampieri, G. -- Analytic discs and extension of CR functions. Compositio Math. 127 (2001), no. 3, 289-295. MR 2002f:32064
  • [Bi65] Bishop, E. -- Differentiable manifolds in complex Euclidean space. Duke Math. J. 32 (1965), 1-21. MR 34:369
  • [Bo91] Boggess, A. -- CR Manifolds and the Tangential Cauchy-Riemann Complex. Studies in Advanced Mathematics. CRC Press. Boca Raton Ann Arbor Boston London 1991. MR 94e:32035
  • [Bo98] Boggess, A. -- The holomorphic extension of $C^k$ CR functions on tube submanifolds. Complex analysis and applications (Warsaw, 1997). Ann. Polon. Math. 70 (1998), 35-42. MR 99j:32010
  • [Bo99] Boggess, A. -- The holomorphic extension of $H^p$-CR functions on tube submanifolds. Proc. Amer. Math. Soc. 127 (1999), no. 5, 1427-1435. MR 99h:32012
  • [BP85] Boggess, A.; Pitts, J. -- CR extension near a point of higher type. Duke Math. J. 52 (1985), 67-102. MR 86k:32012
  • [BP82] Boggess, A.; Polking, J.C. -- Holomorphic extension of CR functions. Duke Math. J. 49, 757-784 (1982). MR 84j:32014
  • [EG01] Eastwood, M.C.; Graham, C.R. -- An Edge-of-the Wedge Theorem for Hypersurface CR Functions. J. Geom. Anal. 11 (2001), no. 4, 589-602. MR 2002i:32033
  • [EG02] Eastwood, M.C.; Graham, C.R. -- Edge of the Wedge Theory in Hypo-Analytic Manifolds, preprint (2001),
  • [HS82] Hanges, N.; Sjöstrand, J. -- Propagation of analyticity for a class of nonmicrocharacteristic operators. Ann. of Math. (2) 116 (1982), no. 3, 559-577. MR 85g:58085
  • [HT83] Hanges, N.; Treves, F. -- Propagation of holomorphic extendability of CR functions. Math. Ann. 263 (1983), no. 2, 157-177. MR 85c:58102
  • [HT78] Hill, C.D.; Taiani, G. -- Families of analytic discs in ${C}^{n}$ with boundaries on a prescribed CR submanifold. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5 (1978), no. 2, 327-380. MR 80c:32023
  • [L00] Lanza de Cristoforis, M. -- Differentiability properties of an abstract autonomous composition operator. J. London Math. Soc. (2), 61 (2000), 923-936. MR 2001c:47069
  • [MMZ02] Meylan, F.; Mir, N.; Zaitsev, D. Analytic regularity of CR-mappings. Math. Res. Lett. 9 (2002), no. 1, 73-93. MR 2003d:32041
  • [Tr90] Trépreau, J.-M. -- Sur la propagation des singularités dans les variétés CR. Bull. Soc. Math. France 118 (1990), no. 4, 403-450. MR 92b:58229
  • [T88] Tumanov, A.E. -- Extension of CR-functions into a wedge from a manifold of finite type. Mat. Sb. (N.S.) 136 (178) (1988), no. 1, 128-139; translation in Math. USSR-Sb. 64 (1989), no. 1, 129-140. MR 89m:32027
  • [T90] Tumanov, A.E. -- Extension of CR-functions into a wedge. Mat. Sb. 181 (1990), no. 7, 951-964; translation in Math. USSR-Sb. 70 (1991), no. 2, 385-398. MR 91f:32010
  • [T93] Tumanov, A.E. -- On the propagation of extendibility of CR functions. Complex analysis and geometry (Trento, 1993), 479-498, Lecture Notes in Pure and Appl. Math., 173, Dekker, New York, 1996. MR 96j:32012
  • [T94] Tumanov, A.E. -- Connections and propagation of analyticity for CR functions. Duke Math. J. 73 (1994), no. 1, 1-24. MR 95i:32025
  • [T95] Tumanov, A.E. -- Propagation of extendibility of CR functions on manifolds with edges. Multidimensional complex analysis and partial differential equations (Sao Carlos, 1995), 259-269, Contemp. Math. 205, Amer. Math. Soc., Providence, RI, 1997. MR 98e:32015
  • [T96] Tumanov, A.E. -- Analytic discs and the extendibility of CR functions. Integral geometry, Radon transforms and complex analysis (Venice, 1996), 123-141, Lecture Notes in Math. 1684, Springer, Berlin, 1998. MR 99f:32024
  • [ZZ01] Zaitsev, D; Zampieri, G. -- Extension of CR-functions on wedges. Math. Ann., to appear.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 32V10, 32V25, 32D15

Retrieve articles in all journals with MSC (2000): 32V10, 32V25, 32D15

Additional Information

Dmitri Zaitsev
Affiliation: School of Mathematics, Trinity College, Dublin 2, Ireland

Giuseppe Zampieri
Affiliation: Dipartimento di Matematica, Università di Padova, via Belzoni 7, 35131 Padova, Italy

Received by editor(s): July 25, 2002
Published electronically: September 22, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society