Plurisubharmonic martingales and barriers in complex quasi-Banach spaces
Annales de l'Institut Fourier, Volume 39 (1989) no. 4, pp. 1007-1060.

We describe the geometrical structure on a complex quasi-Banach space X that is necessay and sufficient for the existence of boundary limits for bounded, X-valued analytic functions on the open unit disc of the complex plane. It is shown that in such spaces, closed bounded subsets have many plurisubharmonic barriers and that bounded upper semi-continuous functions on these sets have arbitrarily small plurisubharmonic perturbations that attain their maximum. This yields a certain representation of the unit ball of X in a nonlinear but plurisubharmonic compactification which in turn implies the convergence of bounded X-valued plurisubharmonic martingales: a result obtained recently by Bu-Schachermayer. A Choquet-type integral representation in terms of Jensen boundary measures is also included. The proofs rely on (analytic) martingale techniques and the results answer various queries of G.A. Edgar. In an appendix, it is established that Hardy martingales embed in analytic functions. Some of these results were established in the Banach space setting in [Ghoussoub-Lindenstraaauss-Maurey, Contemporary Math., vol 85 (1989), 111-130].

Les espaces de Banach complexes (ou plus généralement quasi-normés) qui vérifient la propriété de Rado-Nikodym analytique sont les espaces X tels que toute fonction holomorphe bornée définie dans le disque unité ouvert de et à valeurs dans X admette des limites radiales. Nous décrivons les propriétés géométriques qui caractérisent ces espaces. Nous montrons que dans un tel espace tout ensemble fermé et borné C admet des points barrière-plurisousharmonique et que les fonctions s.c.s. bornées sur C admettent des perturbations par des fonctions plurisousharmoniques (arbitrairement petites sur C) qui atteignent leur maximum sur C. Ces résultats impliquent une représentation de la boule unité de X dans une compactification plurisousharmonique, qui entraîne à son tour la convergence des martingales PSH bornées à valeurs dans X (nous retrouvons ainsi un résultat récent de Bu et Schachermayer). Pour finir, nous présentons un résultat de représentation intégrale au moyen de mesures de Jensen. Dans un appendice, nous établissons que les martingales de Hardy se plongent d’une certaine façon dans les fonctions holomorphes.

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     title = {Plurisubharmonic martingales and barriers in complex {quasi-Banach} spaces},
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Ghoussoub, Nassif; Maurey, Bernard. Plurisubharmonic martingales and barriers in complex quasi-Banach spaces. Annales de l'Institut Fourier, Volume 39 (1989) no. 4, pp. 1007-1060. doi : 10.5802/aif.1198. https://aif.centre-mersenne.org/articles/10.5802/aif.1198/

[A] A. B. Alexandrov, Essays on non locally convex Hardy classes, Lecture notes in Mathematics, Vol. 864, pp. 1-89, Springer-Verlag, Berlin, New York, 1981. | Zbl: 0482.46035

[Bo] J. Bourgain, La propriété de Radon-Nikodym, Publications Mathématiques de l'Université Pierre et Marie Curie, N° 36 (1979).

[Bu] S. Bu, Quelques remarques sur la propriété de Radon-Nikodym analytique, C.R. Acad. Sci. Paris, 306 (1988), 757-760. | MR: 89g:46040 | Zbl: 0651.46026

[BS] S. Bu, W. Schachermayer, Approximation of Jensen Measures by Image Measures under Holomorphic Functions and Applications, (1989) (to appear). | Zbl: 0758.46014

[BD] A. V. Bukhvalov and A. A. Danilevich, Boundary properties of analytic and harmonic functions with values in Banach spaces, Math Zametki, 31 (1982), 203-214 ; English translation, Math. Notes, 31 (1982), 104-110. | MR: 84f:46032 | Zbl: 0496.30029

[C] S. B. Chae, Holomorphy and Calculus in normed spaces, Pure and applied Mathematics, Marcel Dekker Inc., New York and Basel, 1985. | MR: 86j:46044 | Zbl: 0571.46031

[DGT] W. J. Davis, D. J. H. Garling, N. Tomczak-Jaegermann, The complex convexity of complex quasi-normed linear spaces, J. Funct. Anal., 55 (1984), 110-150. | MR: 86b:46032 | Zbl: 0552.46012

[DU] J. Diestel, J. J. Uhl, Vector measures, Math. Surveys, A.M.S. 15 (1977). | MR: 56 #12216 | Zbl: 0369.46039

[D] P. M. Dowling, Representable operators and the analytic Radon-Nikodym property in Banach spaces, Proc. Royal. Irish. Acad., 85A (1985), 143-150. | MR: 87h:46059 | Zbl: 0607.46016

[Dud] R. M. Dudley, Convergence of Baire measure, Studia Math., 27 (1966), 251-268. | MR: 34 #598 | Zbl: 0147.31301

[Du] R. Durett, Brownian motion and Martingales in Analysis, Wadsworth, 1984. | Zbl: 0554.60075

[E1] G. A. Edgar, Complex martingale convergence, Springer-Verlag, Lecture Notes, 1116 (1985), 38-59. | MR: 88a:46013 | Zbl: 0594.60049

[E2] G. A. Edgar, Analytic martingale convergence, J. Funct. Analysis, 69, N° 2 (1986), 268-280. | MR: 88f:46046 | Zbl: 0605.60050

[E3] G. A. Edgar, Extremal Integral Representations, J. Funct. Analysis, Vol. 23, N° 2 (1976), 145-161. | MR: 55 #8753 | Zbl: 0328.46041

[E4] G. A. Edgar, On the Radon-Nikodym Property and Martingale Convergence, Proceedings of Conf. on Vector Space Measures and Applications, Dublin (1977). | Zbl: 0374.46037

[Et] D. O. Etter, Vector-valued Analytic functions, T.A.M.S., 119 (1965), 352-366. | MR: 32 #6186 | Zbl: 0135.16205

[Gam] T. W. Gamelin, Uniform algebras and Jensen measures, Cambridge University Press, Lecture notes series (32) (1978). | MR: 81a:46058 | Zbl: 0418.46042

[Gar] J. B. Garnett, Bounded Analytic Functions, Pure and Applied Math., Academic Press, 1981. | MR: 83g:30037 | Zbl: 0469.30024

[Garl] D. J. H. Garling, On martingales with values in a complex Banach space, Proc. Cambridge. Phil. Soc. (1988) (to appear). | MR: 89g:60160 | Zbl: 0685.46028

[GLM] N. Ghoussoub, J. Lindenstrauss, B. Maurey, Analytic Martingales and Plurisubharmonic Barriers in complex Banach spaces, Contemporary Math. vol. 85, 111-130 (1989). | MR: 90f:46033 | Zbl: 0684.46021

[GM1] N. Ghoussoub, B. Maurey, Gδ-embeddings in Hilbert space, J. Funct. Analysis, 61, N° 1 (1985), 72-97. | MR: 86m:46016 | Zbl: 0565.46011

[GM2] N. Ghoussoub, B. Maurey, Hδ-embeddings in Hilbert space and Optimization on Gδ-sets, Memoirs of A.M.S., 62, N° 349 (1986). | MR: 88i:46022 | Zbl: 0606.46005

[GMS] N. Ghoussoub, B. Maurey, W. Schachermayer, Pluriharmonically Dentable complex Banach spaces... Journal für die reine und angewandte Mathematik, Band 402, 39 (1989), 76-127. | MR: 91f:46024 | Zbl: 0683.46016

[GR] N. Ghoussoub, H. P. Rosenthal, Martingales, Gδ-embeddings and quotients of L1, Math. Annalen, 264 (1983), 321-332. | Zbl: 0511.46017

[HP] U. Haagerup, G. Pisier, Factorization of Analytic Functions with Values in Non-Commutative L1-spaces and Applications, (1988) (to appear). | Zbl: 0821.46074

[JH] R. C. James, A. Ho, The asymptotic norming and Radon-Nikodym properties for Banach spaces, Arkiv fur Matematik, 19 (1981), 53-70. | MR: 82i:46033 | Zbl: 0466.46025

[K1] N. J. Kalton, Plurisubharmonic functions on quasi-Banach spaces, Studia Mathematica, TLXXXIV (1986), 297-324. | MR: 88g:46030 | Zbl: 0625.46021

[K2] N. J. Kalton, Differentiability properties of vector-valued functions, Lecture Notes in Math., Springer-Verlag, 1221 (1985), 141-181. | MR: 88d:46076 | Zbl: 0654.46033

[K3] N. J. Kalton, Analytic functions in non-locally convex spaces and applications, Studia Mathematica, TLXXXIII (1986), 275-303. | MR: 88a:46022 | Zbl: 0634.46038

[K4] N. J. Kalton, Compact Convex Sets and Complex Convexity, Israel J. Math., (1988) (to appear). | Zbl: 0636.46008

[Kh] B. Khaoulani, Un Théorème de représentation intégrale (1989) (In preparation). | Zbl: 0727.46007

[Ko] P. Koosis, Lectures on Hp-spaces. LMS Lecture notes, Cambridge University Press, Cambridge, 1980.

[KPR] N. J. Kalton, N. T. Peck, J. W. Roberts, An F-space Sampler, London Math. Society Lecture Notes 89, Cambridge University Press, 1985. | MR: 87c:46002 | Zbl: 0556.46002

[LS] J. Lindenstrauss, C. Stegall, Examples of separable spaces which do not contain l1 and whose duals are non-separable, Studia Math., 54 (1975), 81-105. | MR: 52 #11543 | Zbl: 0324.46017

[LT] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II : Function Spaces, Erg. Math. Grenzgeb. 97, Berlin-Heidelberg-New York, Springer, 1979. | MR: 81c:46001 | Zbl: 0403.46022

[N] J. Neveu, Discrete parameter martingales, North Holland, 1975. | MR: 53 #6729 | Zbl: 0345.60026

[SSW] W. Schachermayer, A. Sersouri, E. Werner, Israel J. Math., vol. 65 (1989), 225-257. | Zbl: 0686.46011

[St] C. Stegall, Optimization of functions on certain subsets of Banach spaces, Math. Annalen., 236 (1978), 171-176. | MR: 80a:46022 | Zbl: 0365.49006

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