no. 4
Extension and restriction of holomorphic functions
Annales de l'Institut Fourier, Tome 47 (1997) no. 4, pp. 1079-1099.

De fortes pathologies par rapport aux propriétés de croissance peuvent apparaître pour l’extension des fonctions holomorphes d’une sous-variété D ′ d’un domaine D à D tout entier, même dans des cas très simples; les espaces A p (D ′ ):=𝒪(D ′ )∩L p (D ′ ) ne sont, en général, pas tous préservés par extension. De même, la restriction de l’espace de Hilbert A 2 (D) à D ′ , peut avoir une structure très étrange.

Strong pathologies with respect to growth properties can occur for the extension of holomorphic functions from submanifolds D ′ of pseudoconvex domains D to all of D even in quite simple situations; The spaces A p (D ′ ):=𝒪(D ′ )∩L p (D ′ ) are, in general, not at all preserved. Also the image of the Hilbert space A 2 (D) under the restriction to D ′ can have a very strange structure.

@article{AIF_1997__47_4_1079_0,
     author = {Diederich, Klas and Mazzilli, Emmanuel},
     title = {Extension and restriction of holomorphic functions},
     journal = {Annales de l'Institut Fourier},
     pages = {1079--1099},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {47},
     number = {4},
     year = {1997},
     doi = {10.5802/aif.1592},
     zbl = {0881.32005},
     mrnumber = {99d:32012},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1592/}
}
TY  - JOUR
AU  - Diederich, Klas
AU  - Mazzilli, Emmanuel
TI  - Extension and restriction of holomorphic functions
JO  - Annales de l'Institut Fourier
PY  - 1997
SP  - 1079
EP  - 1099
VL  - 47
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1592/
DO  - 10.5802/aif.1592
LA  - en
ID  - AIF_1997__47_4_1079_0
ER  - 
%0 Journal Article
%A Diederich, Klas
%A Mazzilli, Emmanuel
%T Extension and restriction of holomorphic functions
%J Annales de l'Institut Fourier
%D 1997
%P 1079-1099
%V 47
%N 4
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1592/
%R 10.5802/aif.1592
%G en
%F AIF_1997__47_4_1079_0
Diederich, Klas; Mazzilli, Emmanuel. Extension and restriction of holomorphic functions. Annales de l'Institut Fourier, Tome 47 (1997) no. 4, pp. 1079-1099. doi : 10.5802/aif.1592. https://aif.centre-mersenne.org/articles/10.5802/aif.1592/

[1] E. Amar, Extension de fonctions holomorphes et courants, Bull. Sc. Math., 107 (1983), 25-48. | MR | Zbl

[2] E. Amar, Extension de fonctions holomorphes et intégrales singulières, C.R. Acad. Sc. Paris, 299 (1984), 371-374. | MR | Zbl

[3] B. Berndtsson, M. Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier, 32-3 (1982), 91-110. | Numdam | MR | Zbl

[4] A. Cumenge, Extension dans les classes de Hardy de fonctions holomorphes, Ph.D. thesis, Université Paul Sabatier Toulouse, 1980.

[5] J.-P. Demailly, Regularization of closed positive currents and intersection theory, J. Alg. Geom., 1 (1992), 361-409. | MR | Zbl

[6] J.-P. Demailly, Monge-Ampère operators, Lelong numbers and intersection theory, Complex Analysis and Geometry (Ancona, V., Silva, A., eds.), Plenum Press, 1993, pp. 115-193. | Zbl

[7] J.-P. Demailly, Effective bounds for very ample line bundles, Invent. Math., 124 (1996), 243-261. | MR | Zbl

[8] K. Diederich, J.-E. Fornæss, Pseudoconvex domains with real analytic boundary, Ann. Math., 107 (1978), 371-384. | MR | Zbl

[9] K. Diederich, G. Herbort, Extension of holomorphic L2-functions with weighted growth conditions, Nagoya Math. J., 126 (1992), 141-157. | MR | Zbl

[10] K. Diederich, G. Herbort, Geometric and analytic boundary invariants on pseudoconvex domains. Comparison results, J. Geom. Analysis, 3 (1993), 237-267. | MR | Zbl

[11] K. Diederich, G. Herbort, G., Pseudoconvex domains of semiregular type, Contributions to Complex Analysis (Trépreau, H., Skoda, J. M., eds.), Aspects of Mathematics, vol. E 26, Vieweg-Verlag, 1994, pp. 127-162. | MR | Zbl

[12] K. Diederich, G. Herbort, T. Ohsawa, The Bergman kernel on uniformly extendable pseudoconvex domains, Math. Ann., 273 (1986), 471-478. | MR | Zbl

[13] G.-M. Henkin, Continuation of bounded holomorphic functions from submanifolds in general position to strictly pseudoconvex domains, Math. USSR Izvestija, 6 (1972), 536-563. | MR | Zbl

[14] G.-M. Henkin, J. Leiterer, Theory of Functions on Complex Manifolds, Monographs in Mathematics, vol. 79, Birkhäuser Verlag, Basel, 1984. | MR | Zbl

[15] E. Mazzilli, Division et extension des fonctions holomorphes dans les ellipsoides, Ph.D. thesis, Universit Paul Sabatier de Toulouse, 1995.

[16] E. Mazzilli, Extension des fonctions holomorphes, C.R. Acad. Sci. Paris, 321 (1995), 837-841. | MR | Zbl

[17] E. Mazzilli, Extension des fonctions holomorphes dans les pseudo-ellipsoides, to appear in Math. Z., 1996.

[18] T. Ohsawa, On the extension of L2-holomorphic functions II, Publ. RIMS Kyoto Univ., 24 (1988), 265-275. | MR | Zbl

[19] T. Ohsawa, On the extension of L2-holomorphic functions III: negligible weights, Math. Z., 219 (1995), 215-226. | MR | Zbl

[20] T. Ohsawa, K. Takegoshi, On the extension of L2-holomorphic functions, Math. Z., 195 (1987), 197-204. | MR | Zbl

[21] Y. T. Siu, The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi, Preprint, 1995. | Zbl

[22] H. Skoda, Applications de techniques L2 à la théorie des idéaux d'une algèbre de fonctions holomorphes avec poids, Ann. Sci. Ec. Norm. Sup. Paris, 5 (1972), 545-579. | Numdam | MR | Zbl

Cité par Sources :