Function theory and holomorphic maps on symmetric products of planar domains. (English) Zbl 1334.32001
Summary: We show that the \(\overline{\partial}\)-problem is globally regular on a domain in \({\mathbb {C}}^n\), which is the \(n\)-fold symmetric product of a smoothly bounded planar domain. Remmert-Stein type theorems are proved for proper holomorphic maps between equidimensional symmetric products and proper holomorphic maps from Cartesian products to symmetric products. It is shown that proper holomorphic maps between equidimensional symmetric products of smooth planar domains are smooth up to the boundary.
MSC:
| 32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |
| 32W05 | \(\overline\partial\) and \(\overline\partial\)-Neumann operators |
| 32H40 | Boundary regularity of mappings in several complex variables |
Keywords:
symmetric product of smooth planar domains; \(\overline\partial\)-problem; boundary regularityReferences:
| [1] | Agler, J., Young, N.J.: A commutant lifting theorem for a domain in \[{\mathbb{C}}^2\] C2 and spectral interpolation. J. Funct. Anal. 161(2), 452-477 (1999) · Zbl 0943.47005 |
| [2] | Agler, J., Young, N.J.: A Schwarz lemma for the symmetrized bidisc. Bull. Lond. Math. Soc. 33(2), 175-186 (2001) · Zbl 1030.32011 · doi:10.1112/blms/33.2.175 |
| [3] | Agler, J., Young, N.J.: The hyperbolic geometry of the symmetrized bidisc. J. Geom. Anal. 14(3), 375-403 (2004) · Zbl 1055.32010 · doi:10.1007/BF02922097 |
| [4] | Agler, J., Young, N.J.: A model theory for \[\Gamma\] Γ-contractions. J. Oper. Theory 49(1), 45-60 (2003) · Zbl 1019.47013 |
| [5] | Arbarello, E.; Cornalba, M.; Griffith, PA; Harris, J., Geometry of algebraic curves. Vol. I, No. 267 (1985), New York · Zbl 0559.14017 |
| [6] | Bedford, E., Bell, S.: Boundary continuity of proper holomorphic correspondences. Séminaire d’analyse P. Lelong-P. Dolbeault-H. Skoda, années 1983/1984. Lecture Notes in Mathematics, vol. 1198, pp. 47-64. Springer, Berlin (1986) · Zbl 0387.32011 |
| [7] | Bell, S.R., Catlin, D.: Boundary regularity of proper holomorphic mappings. Duke Math. J. 49(2), 385-396 (1982) · Zbl 0475.32011 · doi:10.1215/S0012-7094-82-04924-9 |
| [8] | Bell, S.R., Krantz, S.G.: Smoothness to the boundary of conformal maps, Rocky Mountain. J. Math. 17(1), 23-40 (1987) · Zbl 0626.30005 |
| [9] | Bierstone, E., Milman, P.D.: Semianalytic and subanalytic sets. Inst. Hautes Études Sci. Publ. Math. 67, 5-42 (1988) · Zbl 0674.32002 · doi:10.1007/BF02699126 |
| [10] | Bochnak, J., Coste, M., Roy, M.-F.: Real Algebraic Geometry. Translated from the 1987 French Original. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3, p. 36. Springer, Berlin (1998) · Zbl 0912.14023 |
| [11] | Cartan, H., Quotient d’un espace analytique par un groupe d’automorphismes, 90-102 (1957), Princeton, NJ · Zbl 0084.07202 |
| [12] | Cartan, H., Quotients of complex analytic spaces, 1-15 (1960), Bombay · Zbl 0122.08702 |
| [13] | Chakrabarti, D., Shaw, M.-C.: The Cauchy-Riemann equations on product domains. Math. Ann. 349(4), 977-998 (2011) · Zbl 1223.32023 · doi:10.1007/s00208-010-0547-x |
| [14] | Chakrabarti, D., Shaw, M.-C.: Sobolev regularity of the \[\overline{\partial } \]∂¯-equation on the Hartogs triangle. Math. Ann. 356(1), 241-258 (2013) · Zbl 1276.32027 · doi:10.1007/s00208-012-0840-y |
| [15] | Chaumat, J., Chollet, A.-M.: Régularité höldérienne de l’opérateur \[\overline{\partial } \]∂¯ sur le triangle de Hartogs. Ann. Inst. Fourier (Grenoble) 41(4), 867-882 (1991) · Zbl 0735.32004 · doi:10.5802/aif.1277 |
| [16] | Chaumat, J., Chollet, A.-M.: Noyaux pour résoudre l’equation \[\overline{\partial } \]∂¯ dans des classes ultradifférentiables sur des compacts irréguliers de \[{\mathbb{C}}^nCn\]. Several complex variables (Stockholm, 1987/1988). Mathematical Notes, vol. 38, pp. 205-226. Princeton University Press, Princeton, NJ (1993) · Zbl 0777.32009 |
| [17] | Chen, S-C; Shaw, M-C, Partial differential equations in several complex variables, No. 19 (2001), Providence, RI · Zbl 0963.32001 |
| [18] | Conway, J.B.: Functions of one Complex Variable II. Graduate Texts in Mathematics, vol. 159. Springer, New York (1995) · Zbl 0887.30003 |
| [19] | Costara, C.: The symmetrized bidisc and Lempert’s theorem. Bull. Lond. Math. Soc. 36(5), 656-662 (2004) · Zbl 1065.32006 · doi:10.1112/S0024609304003200 |
| [20] | Costara, C.: The \[2\times 22\]×2 spectral Nevanlinna-Pick problem. J. Lond. Math. Soc. 71(2, 3), 684-702 (2005) · Zbl 1080.32013 · doi:10.1112/S002461070500640X |
| [21] | Dufresnoy, A.: Sur l’opérateur \[d^{\prime \prime }\] d″ et les fonctions différentiables au sens de Whitney. Ann. Inst. Fourier (Grenoble) 29, xvi(1), 229-238 (1979) · Zbl 0387.32011 · doi:10.5802/aif.736 |
| [22] | Diederich, K., Fornaess, J.E.: Boundary regularity of proper holomorphic mappings. Invent. Math. 67(3), 363-384 (1982) · Zbl 0501.32010 · doi:10.1007/BF01398927 |
| [23] | Edigarian, A.: Proper holomorphic self-mappings of the symmetrized bidisc. Ann. Polon. Math. 84(2), 181-184 (2004) · Zbl 1098.32007 · doi:10.4064/ap84-2-8 |
| [24] | Edigarian, A., Zwonek, W.: Geometry of the symmetrized polydisc. Arch. Math. (Basel) 84(4), 364-374 (2005) · Zbl 1068.32012 · doi:10.1007/s00013-004-1183-z |
| [25] | Ehsani, D.: Solution of the \[\overline{\partial } \]∂¯-Neumann problem on a bi-disc. Math. Res. Lett. 10(4), 523-533 (2003) · Zbl 1055.32023 · doi:10.4310/MRL.2003.v10.n4.a11 |
| [26] | Greene, RE; Krantz, SG, Function theory of one complex variable, No. 40 (2002), Providence, RI · Zbl 0988.30001 |
| [27] | Folland, GB; Kohn, JJ, The Neumann problem for the Cauchy-Riemann complex, No. 75 (1972), Princeton, NJ · Zbl 0247.35093 |
| [28] | Gunning, R.C., Rossi, H.: Analytic Functions of Several Complex Variables. AMS Chelsea Publishing, Providence, RI (2009). (Reprint of the 1965 original) · Zbl 1204.01045 |
| [29] | Hörmander, L., An introduction to complex analysis in several variables, No. 7 (1990), Amsterdam · Zbl 0685.32001 |
| [30] | Kohn, J.J.: Global regularity for \[\overline{\partial } \]∂¯ on weakly pseudo-convex manifolds. Trans. Am. Math. Soc. 181, 273-292 (1973) · Zbl 0276.35071 |
| [31] | Kosiński, Ł.: Personal Communication, May 12, 2013. |
| [32] | Kurosh, A.: Higher algebra (Translated from the Russian by George Yankovsky). Mir, Moscow (1988). (Reprint of the 1972 translation) · Zbl 0709.60001 |
| [33] | Magnus, W.: Braid groups: a survey. In: Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973), Lecture Notes in Mathematics, vol. 372, pp. 463-487. Springer, Berlin (1974) · Zbl 0674.32002 |
| [34] | Michel, J., Shaw, M.-C.: The \[\overline{\partial } \]∂¯-problem on domains with piecewise smooth boundaries with applications. Trans. Am. Math. Soc. 351(11), 4365-4380 (1999) · Zbl 0926.35100 · doi:10.1090/S0002-9947-99-02519-2 |
| [35] | Misra, G., Shyam Roy, S., Zhang, G.: Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc. Proc. Am. Math. Soc. 141(7), 2361-2370 (2013) · Zbl 1273.32011 · doi:10.1090/S0002-9939-2013-11514-5 |
| [36] | Narasimhan, R.: Several Complex Variables. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL (1995). (Reprint of the 1971 original) |
| [37] | Olver, P.J.: Classical Invariant Theory. London Mathematical Society Student Texts, vol. 44. Cambridge University Press, Cambridge (1999) · Zbl 0971.13004 · doi:10.1017/CBO9780511623660 |
| [38] | Remmert, R., Stein, K.: Eigentliche holomorphe Abbildungen. Math. Z. 73, 159-189 (1960) · Zbl 0102.06903 · doi:10.1007/BF01162476 |
| [39] | Straube, EJ, Lectures on the \[L^2\] L2-Sobolev theory of the \[\overline{\partial } \]∂¯-Neumann problem (2010), Zürich · Zbl 1247.32003 |
| [40] | Stout, EL, Polynomial convexity, No. 261 (2007), Boston, MA · Zbl 1124.32007 |
| [41] | Tsuji, M.: Potential Theory in Modern Function Theory. Chelsea Publishing Co., New York (1975). (Reprinting of the 1959 original) · Zbl 0322.30001 |
| [42] | Whitney, H.: Complex Analytic Varieties. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1972) · Zbl 0265.32008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
