Cœuré: Fonctions plurisousharmoniques sur les espaces vectoriels topologiques et applications à l'étude des fonctions analytiques

Table of contents for this issue | Previous article | Next article

Gérard Cœuré
Fonctions plurisousharmoniques sur les espaces vectoriels topologiques et applications à l'étude des fonctions analytiques
Annales de l'institut Fourier, 20 no. 1 (1970), p. 361-432, doi: 10.5802/aif.345
Article PDF | Reviews MR 43 #564 | Zbl 0187.39003 | 7 citations in Cedram

Résumé - Abstract

The general properties of plurisubharmonic functions whose set of definition is a finitely-open set of a linear topological space $E$, are proved. If $E$ is assumed locally-convex and quasi-complete, the author generalises the Cauchy measure to ``polycircles"; so, some properties of strictly polar sets in Frechet space are extended in infinitely dimension. The Bremermann characterisation of pseudo-convex sets is extended to a variety $X$ spread over a Banach space $E$. These, when $E$ is separable a new bornological topology, finer than $L$. Nachbin topology is defined on the ring $O_x$ of scalar analytic functions on $X$. So let $(X,Y)$ a scalar extension pair, then $G_x\hookrightarrow G_y$ is a topological isomorphism and $(X,Y)$ is an extension pair for vector valued functions. The spectrum of $G_x$ is studied. The end of this work is a generalisation of Hardy spaces to bounded circular domain in ${\bf C}^n$.

Bibliography

[1] ALEXANDER, Thèse (multigraphiée) (1968).
[2] S. BOCHNER, Classes of holomorphic functions of several variables. Proc. Nat. Ac. Sc. U.S.A. (vol. 46) 720-724. (1960). Zbl 0092.29901
[3] S. BOCHNER et W.T. MARTIN, Several complex variables Princeton (1948). MR 10,366a | Zbl 0041.05205
[4] N. BOURBAKI, Intégration Chap. 5.
[5] M. BRELOT, Allure des fonctions sousharmoniques à la frontière, Math. Nach. (t. 4) 298-307 (1950). MR 13,35b | Zbl 0042.10603
[6] M. BRELOT, G. CHOQUET, J. DENY, Théorie du Potentiel, Séminaire de la Fac. Sc, de Paris (1958).
[7,a] H.J. BREMERMANN, Math. Ann. 173-186 (1958). Zbl 0089.05902
[7,b] Die characterisierung von Regulartästreigebieten (Thèse). Munster 1951.
[8] G. CŒURE, Cr. Ac. Sc. (t. 267) 473-476 et 816 (1968).
[9] G. CŒURE, Cr. Ac. Sc. (t. 267) 440-442 (1968).
[10] G. CŒURE, Cr. Ac. Sc. (t. 262) 177-180 (1966).
[11] G. CŒURE, Cr. Ac. Sc. (t. 264) 287-290 (1967).
[12] A. DOUADY, (Thèse) Ann. de l'Institut Fourier (t. XVI) (1966). Cedram | Zbl 0146.31103
[13] H. FURSTENBERG, A Poisson Formula. Ann. of Math. (t. 77) 325-385 (1966).
[14] I.M. GEL'FAND et N.YA. VILENKIN, Generalised functions (Vol. 4), Acad. Press (1964).
[15] A. GROTHENDIECK, Espaces vectoriels topologiques, Sao-Paulo Univ. (1964).
[16] G. GUNNING et H. ROSSI, Analytic functions of several complex variables, Prentice Hall (1966). Zbl 0141.08601
[17] G.H. HARDY, Proc. London Math. Soc. (T. 14) (1915).
[18] E. HILLE et G. PHILLIPS, Functional analysis and semi-groups, Am. Math. Soc. Publ. (Vol. XXXI) (1957).
[19] S. KAKUTANI, Ann. of Math. (T. 49) (1948).
[20] C. O. KIESELMAN, On entire functions of exponential type. Acta Math. (T. 117) p. 1-35 (1967). Zbl 0152.07602
[21] Y. KUSUNOKI, J. Math. Kyoto Univ, 123-134 (1964). Zbl 0158.12704
[22] P. LELONG, Fonctions plurisousharmoniques, Ann. Ec. Norm. Sup. (t. 62) (p. 301-338) (1945). Numdam | MR 8,271f | Zbl 0061.23205
[23,a] P. LELONG, Séminaires d'analyse de la Fac. Sc. de Paris (1968) n° 71 et 116.
[23,b] “Fonctionnelles analytiques et fonctions entières” Presse de Montréal (1968). Zbl 0194.38801
[24,a] P. LELONG, Fonctions plurisousharmoniques et fonctions analytiques de variables réelles. Ann. Inst. Fourier (t. XI) (1961). Cedram | MR 26 #358 | Zbl 0100.07902
[24,b] Fonctions entières de type exponentielle. Ann. Inst. Fourier. (t. 10) fasc. 2 (1966). Cedram
[25] P. LELONG, Cr. Ac. Sc. (t. 267) p. 916-918 (1968). Zbl 0172.16501
[26,a] P. LELONG, Domaines convexes par rapport aux fonctions plurisousharmoniques. Journal d'Anal. Math. (V. 2) p. 179-207 (1952). MR 14,971c | Zbl 0049.18102
[26,b] Fonctions plurisousharmoniques et formes différentielles positives. Gordon and Beach (1969).
[27] PH. NOVERRAZ, Thèse à paraître Ann. Inst. Fourier (1969).
[28] M.A. ZORN, Characterisation of analytic functions in Banach spaces. Ann. of Math. (t. 12) p. 585-597 (1945). Zbl 0063.08407