Plurisubharmonic martingales and barriers in complex quasi-Banach spaces

Nassif Ghoussoub; Bernard Maurey

Annales de l'institut Fourier (1989)

  • Volume: 39, Issue: 4, page 1007-1060
  • ISSN: 0373-0956

Abstract

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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].

How to cite

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Ghoussoub, Nassif, and Maurey, Bernard. "Plurisubharmonic martingales and barriers in complex quasi-Banach spaces." Annales de l'institut Fourier 39.4 (1989): 1007-1060. <http://eudml.org/doc/74855>.

@article{Ghoussoub1989,
abstract = {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].},
author = {Ghoussoub, Nassif, Maurey, Bernard},
journal = {Annales de l'institut Fourier},
keywords = {Brownian motion; Choquet-type integral representation; analytic martingale techniques; geometrical structure on a complex quasi-Banach space; plurisubharmonic barriers; bounded upper semi-continuous functions; plurisubharmonic perturbations; plurisubharmonic martingales; Jensen boundary measures; Hardy martingales},
language = {eng},
number = {4},
pages = {1007-1060},
publisher = {Association des Annales de l'Institut Fourier},
title = {Plurisubharmonic martingales and barriers in complex quasi-Banach spaces},
url = {http://eudml.org/doc/74855},
volume = {39},
year = {1989},
}

TY - JOUR
AU - Ghoussoub, Nassif
AU - Maurey, Bernard
TI - Plurisubharmonic martingales and barriers in complex quasi-Banach spaces
JO - Annales de l'institut Fourier
PY - 1989
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 4
SP - 1007
EP - 1060
AB - 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].
LA - eng
KW - Brownian motion; Choquet-type integral representation; analytic martingale techniques; geometrical structure on a complex quasi-Banach space; plurisubharmonic barriers; bounded upper semi-continuous functions; plurisubharmonic perturbations; plurisubharmonic martingales; Jensen boundary measures; Hardy martingales
UR - http://eudml.org/doc/74855
ER -

References

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  1. [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. Zbl0482.46035
  2. [Bo] J. BOURGAIN, La propriété de Radon-Nikodym, Publications Mathématiques de l'Université Pierre et Marie Curie, N° 36 (1979). 
  3. [Bu] S. BU, Quelques remarques sur la propriété de Radon-Nikodym analytique, C.R. Acad. Sci. Paris, 306 (1988), 757-760. Zbl0651.46026MR89g:46040
  4. [BS] S. BU, W. SCHACHERMAYER, Approximation of Jensen Measures by Image Measures under Holomorphic Functions and Applications, (1989) (to appear). Zbl0758.46014
  5. [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. Zbl0496.30029MR84f:46032
  6. [C] S. B. CHAE, Holomorphy and Calculus in normed spaces, Pure and applied Mathematics, Marcel Dekker Inc., New York and Basel, 1985. Zbl0571.46031MR86j:46044
  7. [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. Zbl0552.46012MR86b:46032
  8. [DU] J. DIESTEL, J. J. UHL, Vector measures, Math. Surveys, A.M.S. 15 (1977). Zbl0369.46039MR56 #12216
  9. [D] P. M. DOWLING, Representable operators and the analytic Radon-Nikodym property in Banach spaces, Proc. Royal. Irish. Acad., 85A (1985), 143-150. Zbl0607.46016MR87h:46059
  10. [Dud] R. M. DUDLEY, Convergence of Baire measure, Studia Math., 27 (1966), 251-268. Zbl0147.31301MR34 #598
  11. [Du] R. DURETT, Brownian motion and Martingales in Analysis, Wadsworth, 1984. Zbl0554.60075
  12. [E1] G. A. EDGAR, Complex martingale convergence, Springer-Verlag, Lecture Notes, 1116 (1985), 38-59. Zbl0594.60049MR88a:46013
  13. [E2] G. A. EDGAR, Analytic martingale convergence, J. Funct. Analysis, 69, N° 2 (1986), 268-280. Zbl0605.60050MR88f:46046
  14. [E3] G. A. EDGAR, Extremal Integral Representations, J. Funct. Analysis, Vol. 23, N° 2 (1976), 145-161. Zbl0328.46041MR55 #8753
  15. [E4] G. A. EDGAR, On the Radon-Nikodym Property and Martingale Convergence, Proceedings of Conf. on Vector Space Measures and Applications, Dublin (1977). Zbl0374.46037
  16. [Et] D. O. ETTER, Vector-valued Analytic functions, T.A.M.S., 119 (1965), 352-366. Zbl0135.16205MR32 #6186
  17. [Gam] T. W. GAMELIN, Uniform algebras and Jensen measures, Cambridge University Press, Lecture notes series (32) (1978). Zbl0418.46042MR81a:46058
  18. [Gar] J. B. GARNETT, Bounded Analytic Functions, Pure and Applied Math., Academic Press, 1981. Zbl0469.30024MR83g:30037
  19. [Garl] D. J. H. GARLING, On martingales with values in a complex Banach space, Proc. Cambridge. Phil. Soc. (1988) (to appear). Zbl0685.46028MR89g:60160
  20. [GLM] N. GHOUSSOUB, J. LINDENSTRAUSS, B. MAUREY, Analytic Martingales and Plurisubharmonic Barriers in complex Banach spaces, Contemporary Math. vol. 85, 111-130 (1989). Zbl0684.46021MR90f:46033
  21. [GM1] N. GHOUSSOUB, B. MAUREY, Gδ-embeddings in Hilbert space, J. Funct. Analysis, 61, N° 1 (1985), 72-97. Zbl0565.46011MR86m:46016
  22. [GM2] N. GHOUSSOUB, B. MAUREY, Hδ-embeddings in Hilbert space and Optimization on Gδ-sets, Memoirs of A.M.S., 62, N° 349 (1986). Zbl0606.46005MR88i:46022
  23. [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. Zbl0683.46016MR91f:46024
  24. [GR] N. GHOUSSOUB, H. P. ROSENTHAL, Martingales, Gδ-embeddings and quotients of L1, Math. Annalen, 264 (1983), 321-332. Zbl0511.46017
  25. [HP] U. HAAGERUP, G. PISIER, Factorization of Analytic Functions with Values in Non-Commutative L1-spaces and Applications, (1988) (to appear). Zbl0821.46074
  26. [JH] R. C. JAMES, A. HO, The asymptotic norming and Radon-Nikodym properties for Banach spaces, Arkiv fur Matematik, 19 (1981), 53-70. Zbl0466.46025MR82i:46033
  27. [K1] N. J. KALTON, Plurisubharmonic functions on quasi-Banach spaces, Studia Mathematica, TLXXXIV (1986), 297-324. Zbl0625.46021MR88g:46030
  28. [K2] N. J. KALTON, Differentiability properties of vector-valued functions, Lecture Notes in Math., Springer-Verlag, 1221 (1985), 141-181. Zbl0654.46033MR88d:46076
  29. [K3] N. J. KALTON, Analytic functions in non-locally convex spaces and applications, Studia Mathematica, TLXXXIII (1986), 275-303. Zbl0634.46038MR88a:46022
  30. [K4] N. J. KALTON, Compact Convex Sets and Complex Convexity, Israel J. Math., (1988) (to appear). Zbl0636.46008
  31. [Kh] B. KHAOULANI, Un Théorème de représentation intégrale (1989) (In preparation). Zbl0727.46007
  32. [Ko] P. KOOSIS, Lectures on Hp-spaces. LMS Lecture notes, Cambridge University Press, Cambridge, 1980. 
  33. [KPR] N. J. KALTON, N. T. PECK, J. W. ROBERTS, An F-space Sampler, London Math. Society Lecture Notes 89, Cambridge University Press, 1985. Zbl0556.46002MR87c:46002
  34. [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. Zbl0324.46017MR52 #11543
  35. [LT] J. LINDENSTRAUSS, L. TZAFRIRI, Classical Banach Spaces II : Function Spaces, Erg. Math. Grenzgeb. 97, Berlin-Heidelberg-New York, Springer, 1979. Zbl0403.46022MR81c:46001
  36. [N] J. NEVEU, Discrete parameter martingales, North Holland, 1975. Zbl0345.60026MR53 #6729
  37. [SSW] W. SCHACHERMAYER, A. SERSOURI, E. WERNER, Israel J. Math., vol. 65 (1989), 225-257. Zbl0686.46011
  38. [St] C. STEGALL, Optimization of functions on certain subsets of Banach spaces, Math. Annalen., 236 (1978), 171-176. Zbl0365.49006MR80a:46022

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