Duality for some large spaces of analytic functions

Abstract

We characterize the duals and biduals of the Lp-analogues Nαp of the standard Nevanlinna classes Nα, α ≥ -1 and 1 ≤ p < ∞. We adopt the convention to take N-1p to be the classical Smirnov class N+ for p = 1, and the Hardy-Orlicz space LHP (= (Log+H)p) for 1 < p < ∞. Our results generalize and unify earlier characterizations obtained by Eoff for α = 0 and α = -1, and by Yanigahara for the Smirnov class. Each Nαp is a complete metrizable topological vector space (in fact, even an algebra); it fails to be locally bounded and locally convex but admits a separating dual. Its bidual will be identified with a specific nuclear power series space of finite type; this turns out to be the 'Fréchet envelope' of Nαp as well. The generating sequence of this power series space is of the form (nθ)n∈ℕ for some 0 < θ < 1. For example, the θs in the interval (1/2, 1) correspond in a bijective fashion to the Nevanlinna classes Nα, α > -1, whereas the θs in the interval (0, 1/2) correspond bijectively to the Hardy-Orlicz spaces LHP, 1 < p