Plurisubharmonic martingales and barriers in complex quasi-Banach spaces

Ghoussoub, Nassif; Maurey, Bernard

Annales de l'Institut Fourier, Tome 39 (1989) no. 4, pp. 1007-1060.

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 $ℂ$ 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.

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].

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