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 -

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