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

Zaitsev, Dmitri; Zampieri, Giuseppe

doi:10.1090/S0002-9947-03-03356-7

Extension of CR-functions into weighted wedges through families of nonsmooth analytic discs
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by Dmitri Zaitsev and Giuseppe Zampieri

Trans. Amer. Math. Soc. 356 (2004), 1443-1462

DOI: https://doi.org/10.1090/S0002-9947-03-03356-7

Published electronically: September 22, 2003

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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

R. A. Aĭrapetjan and G. M. Henkin, Analytic continuation of CR-functions across the “edge of the wedge”, Dokl. Akad. Nauk SSSR 259 (1981), no. 4, 777–781 (Russian). MR 624846
M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999. MR 1668103, DOI 10.1515/9781400883967
M. S. Baouendi, L. P. Rothschild, and J.-M. Trépreau, On the geometry of analytic discs attached to real manifolds, J. Differential Geom. 39 (1994), no. 2, 379–405. MR 1267896
M. Tursunov, A system of differential equations with singular coefficients, Dokl. Akad. Nauk Tadzhik. SSR 23 (1980), no. 11, 620–624 (Russian, with Tajiki summary). MR 612261
Luca Baracco and Giuseppe Zampieri, Analytic discs and extension of CR functions, Compositio Math. 127 (2001), no. 3, 289–295. MR 1845039, DOI 10.1023/A:1017581000515
Errett Bishop, Differentiable manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1–21. MR 200476
Albert Boggess, CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1991. MR 1211412
Al Boggess, The holomorphic extension of $C^k$ CR functions on tube submanifolds, Ann. Polon. Math. 70 (1998), 35–42. Complex analysis and applications (Warsaw, 1997). MR 1668714, DOI 10.4064/ap-70-1-35-42
Al Boggess, The holomorphic extension of $H^p$-CR functions on tube submanifolds, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1427–1435. MR 1600104, DOI 10.1090/S0002-9939-99-04828-5
A. Boggess and J. Pitts, CR extension near a point of higher type, Duke Math. J. 52 (1985), no. 1, 67–102. MR 791293, DOI 10.1215/S0012-7094-85-05206-8
Xavier Gómez-Mont, Locally determined ideals in analytic spaces, An. Inst. Mat. Univ. Nac. Autónoma México 20 (1980), no. 2, 177–189 (1982) (Spanish). MR 687851
Michael G. Eastwood and C. Robin Graham, An edge-of-the-wedge theorem for hypersurface CR functions, J. Geom. Anal. 11 (2001), no. 4, 589–602. MR 1861297, DOI 10.1007/BF02930756
Eastwood, M.C.; Graham, C.R. — Edge of the Wedge Theory in Hypo-Analytic Manifolds, preprint (2001), http://arXiv.org/abs/math.CV/0107183.
Nicholas Hanges and Johannes Sjöstrand, Propagation of analyticity for a class of nonmicrocharacteristic operators, Ann. of Math. (2) 116 (1982), no. 3, 559–577. MR 678481, DOI 10.2307/2007023
Nicholas Hanges and François Trèves, Propagation of holomorphic extendability of CR functions, Math. Ann. 263 (1983), no. 2, 157–177. MR 698000, DOI 10.1007/BF01456878
C. Denson Hill and Geraldine Taiani, Families of analytic discs in $\textbf {C}^{n}$ with boundaries on a prescribed CR submanifold, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978), no. 2, 327–380. MR 501906
Massimo Lanza de Cristoforis, Differentiability properties of an abstract autonomous composition operator, J. London Math. Soc. (2) 61 (2000), no. 3, 923–936. MR 1766115, DOI 10.1112/S0024610700008802
Francine Meylan, Nordine Mir, and Dmitri Zaitsev, Analytic regularity of CR-mappings, Math. Res. Lett. 9 (2002), no. 1, 73–93. MR 1892315, DOI 10.4310/MRL.2002.v9.n1.a6
J.-M. Trépreau, Sur la propagation des singularités dans les variétés CR, Bull. Soc. Math. France 118 (1990), no. 4, 403–450 (French, with English summary). MR 1090408
A. E. Tumanov, Extension of CR-functions into a wedge from a manifold of finite type, Mat. Sb. (N.S.) 136(178) (1988), no. 1, 128–139 (Russian); English transl., Math. USSR-Sb. 64 (1989), no. 1, 129–140. MR 945904, DOI 10.1070/SM1989v064n01ABEH003298
A. E. Tumanov, Extension of CR-functions into a wedge, Mat. Sb. 181 (1990), no. 7, 951–964 (Russian); English transl., Math. USSR-Sb. 70 (1991), no. 2, 385–398. MR 1070489, DOI 10.1070/SM1991v070n02ABEH001258
A. Tumanov, On the propagation of extendibility of CR functions, Complex analysis and geometry (Trento, 1993) Lecture Notes in Pure and Appl. Math., vol. 173, Dekker, New York, 1996, pp. 479–498. MR 1365986
A. Tumanov, Connections and propagation of analyticity for CR functions, Duke Math. J. 73 (1994), no. 1, 1–24. MR 1257276, DOI 10.1215/S0012-7094-94-07301-8
A. Tumanov, Propagation of extendibility of CR functions on manifolds with edges, Multidimensional complex analysis and partial differential equations (São Carlos, 1995) Contemp. Math., vol. 205, Amer. Math. Soc., Providence, RI, 1997, pp. 259–269. MR 1447230, DOI 10.1090/conm/205/02669
Alexander Tumanov, Analytic discs and the extendibility of CR functions, Integral geometry, Radon transforms and complex analysis (Venice, 1996) Lecture Notes in Math., vol. 1684, Springer, Berlin, 1998, pp. 123–141. MR 1635614, DOI 10.1007/BFb0096093
Zaitsev, D; Zampieri, G. — Extension of CR-functions on wedges. Math. Ann., to appear.