Peak points for pseudoconvex domains: A survey. (English) Zbl 1196.32020
The paper under review is an excellent survey article discussing various aspects of the existence of local and/or global peak functions in \(\mathcal C^\infty\)-smooth pseudoconvex domains. Some of the presented results have not been published elsewhere.
Contents: 5mm
Contents: 5mm
- 1.
- Introduction;
- 2.
- Background information and the strictly pseudoconvex case;
- 3.
- Finite type, the Kohn-Nirenberg domain, and strict type;
- 4.
- The sector method;
- 5.
- Alternative constructions of peak functions;
- 6.
- Domains of finite type in \(\mathbb C^n\);
- 7.
- Miscellany.
Reviewer: Marek Jarnicki (Kraków)
MSC:
| 32T40 | Peak functions |
| 32-02 | Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces |
References:
| [1] | Basener, R.: Peak points, barriers and pseudoconvex boundary points. Proc. Am. Math. Soc. 65(1), 89–92 (1977) · Zbl 0392.32003 · doi:10.1090/S0002-9939-1977-0466633-9 |
| [2] | Bedford, E., Fornæss, J.E.: A construction of peak functions on weakly pseudoconvex domains. Ann. Math. (2) 107(3), 555–568 (1978) · Zbl 0392.32004 · doi:10.2307/1971128 |
| [3] | Bedford, E., Fornæss, J.E.: Biholomorphic maps of weakly pseudoconvex domains. Duke Math. J. 45(4), 711–719 (1978) · Zbl 0401.32006 · doi:10.1215/S0012-7094-78-04533-7 |
| [4] | Bell, S.: Differentiability of the Bergman kernel and pseudolocal estimates. Math. Z. 192(3), 467–472 (1986) · Zbl 0594.32025 · doi:10.1007/BF01164021 |
| [5] | Bharali, G., Stensønes, B.: Plurisubharmonic polynomials and bumping, arXiv:0709.3993v2 . Math. Z. (2008, to appear) · Zbl 1185.32025 |
| [6] | Bloom, T.: \(\mathcal{C}^{\infty}\) peak functions for pseudoconvex domains of strict type. Duke Math. J. 45(1), 133–147 (1978) · Zbl 0376.32014 · doi:10.1215/S0012-7094-78-04510-6 |
| [7] | Boas, H.: Extension of Kerzman’s theorem on differentiability of the Bergman kernel function. Indiana Univ. Math. J. 36(3), 495–499 (1987) · Zbl 0628.32030 · doi:10.1512/iumj.1987.36.36027 |
| [8] | Browder, A.: Introduction to Function Algebras. Benjamin, Elmsford (1969) · Zbl 0199.46103 |
| [9] | Catlin, D.: Boundary invariants of pseudoconvex domains. Ann. Math. (2) 120(3), 529–586 (1984) · Zbl 0583.32048 · doi:10.2307/1971087 |
| [10] | Chaumat, J., Chollet, A.M.: Ensemble de zéros, ensembles pics et d’interpolation pour A(D). Colloque d’Analyse Harmonique et Complexe, 5 p. (not consecutively paged). Univ. Aix-Marseille I, Marseille (1977) · Zbl 0473.32009 |
| [11] | Chen, S.C.: A counterexample to the differentiability of the Bergman kernel function. Proc. Am. Math. Soc. 124(6), 1807–1810 (1996) · Zbl 0865.32016 · doi:10.1090/S0002-9939-96-03290-X |
| [12] | Cho, S.: A construction of peak functions on locally convex domains in C n . Nagoya Math. J. 140, 167–176 (1995) · Zbl 0848.32010 |
| [13] | Cho, S.: Peak function and its applications. J. Korean Math. Soc. 33(2), 399–411 (1996) · Zbl 0870.32005 |
| [14] | D’Angelo, J.: Several Complex Variables and the Geometry of Real Hypersurfaces. Studies in Advanced Mathematics. CRC, Boca Raton (1993) |
| [15] | Diederich, K., Fornæss, J.E.: Pseudoconvex domains: an example with nontrivial Nebenhülle. Math. Ann. 225(3), 275–292 (1977) · doi:10.1007/BF01425243 |
| [16] | Diederich, K., Fornæss, J.E.: Pseudoconvex domains: existence of Stein neighborhoods. Duke Math. J. 44(3), 641–662 (1977) · Zbl 0381.32014 · doi:10.1215/S0012-7094-77-04427-1 |
| [17] | Diederich, K., Fornæss, J.E.: Pseudoconvex domains with real-analytic boundary. Ann. Math. (2) 107(2), 371–384 (1978) · Zbl 0378.32014 · doi:10.2307/1971120 |
| [18] | Diederich, K., Fornæss, J.E.: Support functions for convex domains of finite type. Math. Z. 230(1), 145–164 (1999) · Zbl 1045.32016 · doi:10.1007/PL00004683 |
| [19] | Diederich, K., Fornæss, J.E.: Lineally convex domains of finite type: holomorphic support functions. Manuscr. Math. 112(4), 403–431 (2003) · Zbl 1066.32016 · doi:10.1007/s00229-003-0418-9 |
| [20] | Diederich, K., Herbort, G.: Pseudoconvex domains of semiregular type. In: Contributions to Complex Analysis and Analytic Geometry. Aspects Math., vol. E26, pp. 127–161. Vieweg, Wiesbaden (1994) · Zbl 0845.32019 |
| [21] | Fornæss, J.E.: Peak points on weakly pseudoconvex domains. Math. Ann. 227(2), 173–175 (1977) · Zbl 0346.32026 · doi:10.1007/BF01350193 |
| [22] | Fornæss, J.E.: Sup-norm estimates for \(\overline{\partial}\) in C 2. Ann. Math. (2) 123(2), 335–345 (1986) · Zbl 0589.32038 · doi:10.2307/1971275 |
| [23] | Fornæss, J.E., Krantz, S.G.: Continuously varying peaking functions. Pac. J. Math. 83(2), 341–347 (1979) · Zbl 0425.46039 |
| [24] | Fornæss, J.E., McNeal, J.: A construction of peak functions on some finite type domains. Am. J. Math. 116(3), 737–755 (1994) · Zbl 0809.32005 · doi:10.2307/2374998 |
| [25] | Fornæss, J.E., Rea, C.: Local holomorphic extendability and nonextendability of CR-functions on smooth boundaries. Ann. Sc. Norm. Super Pisa Cl. Sci. (4) 12(3), 491–502 (1985) · Zbl 0587.32035 |
| [26] | Fornæss, J.E., Sibony, N.: Construction of P.S.H. functions on weakly pseudoconvex domains. Duke Math. J. 58(3), 633–655 (1989) · Zbl 0679.32017 · doi:10.1215/S0012-7094-89-05830-4 |
| [27] | Fornæss, J.E., Stensønes, B.: Lectures on Counterexamples in Several Complex Variables. Math. Notes, vol. 33. Princeton Univ. Press, Princeton (1987) · Zbl 1126.32001 |
| [28] | Freeman, M.: Local complex foliation of real submanifolds. Math. Ann. 209, 1–30 (1974) · doi:10.1007/BF01432883 |
| [29] | Gamelin, T.W.: Peak points for algebras on circled sets. Math. Ann. 238(2), 131–139 (1978) · doi:10.1007/BF01424770 |
| [30] | Gay, R., Sebbar, A.: Division et extension dans l’algèbre A {\(\Omega\)}) d’un ouvert pseudo-convexe à bord lisse de C n . Math. Z. 189(3), 421–447 (1985) · doi:10.1007/BF01164163 |
| [31] | Graham, I.: Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in C n with smooth boundary. Trans. Am. Math. Soc. 207, 219–240 (1975) · Zbl 0305.32011 |
| [32] | Hakim, M., Sibony, N.: Frontière de Ŝilov et spectre de \(A(\overline{D})\) pour des domaines faiblement pseudoconvexes. C. R. Acad. Sci. Paris Sér. A–B 281(22), A959–A962 (1975) · Zbl 0324.46058 |
| [33] | Hakim, M., Sibony, N.: Quelques conditions pour l’existence de fonctions pics dans des domaines pseudoconvexes. Duke Math. J. 44(2), 399–406 (1977) · Zbl 0354.32028 · doi:10.1215/S0012-7094-77-04418-0 |
| [34] | Herbort, G.: Invariant metrics and peak functions on pseudoconvex domains of homogeneous finite diagonal type. Math. Z. 209(2), 223–243 (1992) · doi:10.1007/BF02570831 |
| [35] | Herbort, G.: Localization lemmas for the Bergman metric at plurisubharmonic peak points. Nagoya Math. J. 171, 107–125 (2003) · Zbl 1045.32009 |
| [36] | Iordan, A.: Pseudoconvex domains with peak functions at each point of the boundary. Pac. J. Math. 133(2), 277–287 (1988) · Zbl 0619.32012 |
| [37] | Kerzman, N.: The Bergman kernel function. Differentiability at the boundary. Math. Ann. 195, 149–158 (1972) · doi:10.1007/BF01419622 |
| [38] | Kohn, J.J.: Boundary behavior of \(\overline{\partial}\) on weakly pseudo-convex manifolds of dimension two. J. Differ. Geom. 6, 523–542 (1972) · Zbl 0256.35060 |
| [39] | Kohn, J.J.: Global regularity for \(\overline{\partial}\) on weakly pseudo-convex manifolds. Trans. Am. Math. Soc. 181, 273–292 (1973) · Zbl 0276.35071 |
| [40] | Kohn, J.J.: Methods of partial differential equations in complex analysis. In: Several Complex Variables, Part 1. Proc. Sympos. Pure Math., vol. XXX, Williams Coll., Williamstown, MA, 1975, pp. 215–237. Am. Math. Soc., Providence (1977) |
| [41] | Kohn, J.J., Nirenberg, L.: A pseudo-convex domain not admitting a holomorphic support function. Math. Ann. 201, 265–268 (1973) · Zbl 0248.32013 · doi:10.1007/BF01428194 |
| [42] | Kolář, M.: Convexifiability and supporting functions in C 2. Math. Res. Lett. 2(4), 505–513 (1995) · Zbl 0846.32011 |
| [43] | Kolář, M.: Peak functions on convex domains. In: The Proceedings of the 19th Winter School ”Geometry and Physics” (Srní, 1999). Rend. Circ. Mat. Palermo (2) Suppl. No. 63, pp. 103–112 (2000) · Zbl 0976.32018 |
| [44] | Laszlo, G.: Peak functions on finite type domains in C 2. Dissertation, Oklahoma State University (2000) |
| [45] | Levi, E.E.: Studii sui punti singolari essenziali delle funzioni analitiche di due o più variabili complesse. Ann. Mat. Pura Appl. 17, 61–87 (1910) · JFM 41.0487.01 |
| [46] | Levi, E.E.: Sulle ipersuperficie dello spazio a 4 dimensioni che possono essere frontiera del campo di esistenza di una funzione analitica di due variabili complesse. Ann. Mat. Pura Appl. 18, 69–79 (1911) · JFM 42.0449.02 |
| [47] | McNeal, J.: Lower bounds on the Bergman metric near a point of finite type. Ann. Math. (2) 136(2), 339–360 (1992) · Zbl 0764.32006 · doi:10.2307/2946608 |
| [48] | Noell, A.: Properties of peak sets in weakly pseudoconvex boundaries in C 2. Math. Z. 186, 99–116 (1984) · doi:10.1007/BF01215494 |
| [49] | Noell, A.: Peak points in boundaries not of finite type. Pac. J. Math. 123(2), 385–390 (1986) · Zbl 0563.32006 |
| [50] | Noell, A.: Interpolation from curves in pseudoconvex boundaries. Mich. Math. J. 37(2), 275–281 (1990) · Zbl 0719.32013 · doi:10.1307/mmj/1029004134 |
| [51] | Noell, A.: Peak functions for pseudoconvex domains in C n . In: Several Complex Variables, Stockholm, 1987/1988. Math. Notes, vol. 38, pp. 529–541. Princeton Univ. Press, Princeton (1993) · Zbl 0774.32011 |
| [52] | Noell, A., Stensønes, B.: Proper holomorphic maps from weakly pseudoconvex domains. Duke Math. J. 60(2), 363–388 (1990) · Zbl 0716.32017 |
| [53] | Pflug, P.: Über polynomiale Funktionen auf Holomorphiegebieten. Math. Z. 139, 133–139 (1974) · Zbl 0287.32011 |
| [54] | Pflug, P.: Quadratintegrable holomorphe Funktionen und die Serre-Vermutung. Math. Ann. 216, 285–288 (1975) · Zbl 0294.32009 |
| [55] | Range, R.M.: Hölder estimates for \(\overline{\partial}\) on convex domains in C 2 with real analytic boundary. In: Several Complex Variables, Part 2. Proc. Sympos. Pure Math., vol. XXX, Williams Coll., Williamstown, MA, 1975, pp. 31–33. Am. Math. Soc., Providence (1977) |
| [56] | Range, R.M.: The Carathéodory metric and holomorphic maps on a class of weakly pseudoconvex domains. Pac. J. Math. 78(1), 173–189 (1978) · Zbl 0396.32005 |
| [57] | Range, R.M.: On Hölder estimates for \(\overline{\partial}u=f\) on weakly pseudoconvex domains. In: Several Complex Variables, Cortona, 1976/1977, pp. 247–267. Scuola Norm. Sup. Pisa, Pisa (1978) |
| [58] | Rossi, H.: Holomorphically convex sets in several complex variables. Ann. Math. (2) 74, 470–493 (1961) · Zbl 0107.28601 |
| [59] | Sibony, N.: Un exemple de domaine pseudoconvexe régulier où l’équation \(\bar{\partial}u=f\) n’admet pas de solution bornée pour f bornée. Invent. Math. 62(2), 235–242 (1980/81) |
| [60] | Sibony, N.: Some aspects of weakly pseudoconvex domains, Part 1. In: Several Complex Variables and Complex Geometry, Santa Cruz, CA, 1989, Proc. Sympos. Pure Math., vol. 52, pp. 199–231. Am. Math. Soc., Providence (1991) |
| [61] | Verdera, J.: A remark on zero and peak sets on weakly pseudoconvex domains. Bull. Lond. Math. Soc. 16(4), 411–412 (1984) · Zbl 0578.32025 |
| [62] | Weinstock, B.: Zero-sets of continuous holomorphic functions on the boundary of a strongly pseudoconvex domain. J. Lond. Math. Soc. (2) 18(3), 484–488 (1978) · Zbl 0413.32008 |
| [63] | Yu, J.: Geometric analysis on weakly pseudoconvex domains. Dissertation, Washington University, St. Louis (1993) |
| [64] | Yu, J.: Multitypes of convex domains. Indiana Univ. Math. J. 41(3), 837–849 (1992) · Zbl 0759.32009 |
| [65] | Yu, J.: Peak functions on weakly pseudoconvex domains. Indiana Univ. Math. J. 43(4), 1271–1295 (1994) · Zbl 0828.32003 |
| [66] | Yu, J.: Weighted boundary limits of the generalized Kobayashi-Royden metrics on weakly pseudoconvex domains. Trans. Am. Math. Soc. 347(2), 587–614 (1995) · Zbl 0814.32006 |
| [67] | Yu, J.: A counterexample to the existence of peaking functions. Proc. Am. Math. Soc. 125(8), 2385–2390 (1997) · Zbl 0896.32007 · doi:10.1090/S0002-9939-97-03936-1 |
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