The Plurisubharmonic Dentability and the Analytic Radon-Nikodym Property for Bounded Subsets in Complex Banach Spaces

Abstract

Let X be a complex Banach space. Following [2], X is said to have the analytic Radon- Nikodym property if every uniformly bounded analytic function from the open unit disk of C with values in X, f: D → X has radial limits almost everywhere on the torus T = {e^iθ : θ ∈ [0, 2π[} in X, which means that for almost all θ ∈ [0, 2π[, lim_r↑1 f(re^iθ) exists. An upper semi- continuous function ø : X → R is plurisubharmonic ( see [3] ) if, for every $x,y \in X,f(x) \leqslant \int_0^{2\pi } {f(x + e^{i\theta } y)} \tfrac{{d\theta }} {{2\pi }}.$ We shall denote by PSH ₀(X) the the set of all Lipschitz plurisubharmonic functions f on X satisfying f(0) = 0. For f, g ∈ PSH ₀(X) define d(f, g) as the Lipschitz constant ‖f−g ‖_Lip of f−g. PSH ₀ X equipped with the metric d becomes a complete metric space. A sequence of functions (f _n)_n≥0 in L ¹([0, 2π[^N, X) is called an X-valued analytic martingale if f ₀ ≡ x ₀ ∈ X, f _n ∈ L ¹([0, 2π[ⁿ, X) and for every (θ ₁, θ ₂, … θ _n-1) ∈ [0, 2π[^n-1, f _n(θ ₁, θ ₂, … θ _n) = f _n-1(θ ₁, θ ₂, … θ _n-1) + d _n(θ ₁, θ ₂, … θ _n-1)e^iθn. Among other known characterizations of the analytic Radon-Nikodym property, we have the following