PROJECTIVE DESCRIPTION OF WEIGHTED (LF)-SPACES OF HOLOMORPHIC FUNCTIONS ON THE DISC

The topology of certain weighted inductive limits of Fréchet spaces of holomorphic functions on the unit disc can be described by means of weighted sup-seminorms in case the weights are radial and satisfy certain natural assumptions due to Lusky; in the sense of Shields and Williams the weights have to be normal. It turns out that no assumption on the (double) sequence of normal weights is necessary for the topological projective description in the case of o-growth conditions. For O-growth conditions, we give a necessary and sufficient condition (in terms of associated weights) for projective description in the case of (LB)-spaces and normal weights. This last result is related to a theorem of Mattila, Saksman and Taskinen.AMS 2000 Mathematics subject classification: Primary 46E10. Secondary 30H05; 46A13; 46M40

DUALITY FOR SOME LARGE SPACES OF ANALYTIC FUNCTIONS

We characterize the duals and biduals of the $L^p$-analogues $\mathcal{N}_\alpha^p$ of the standard Nevanlinna classes $\mathcal{N}_\alpha$, $\alpha\ge-1$ and $1\le p\lt \infty$. We adopt the convention to take $\mathcal{N}_{-1}^p$ to be the classical Smirnov class $\mathcal{N}^+$ for $p=1$, and the Hardy–Orlicz space $LH^p$ $(=(\text{Log}^+H)^p)$ for $1\lt p\lt\infty$. Our results generalize and unify earlier characterizations obtained by Eoff for $\alpha=0$ and $\alpha=-1$, and by Yanigahara for the Smirnov class.Each $\mathcal{N}_\alpha^p$ 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 $\mathcal{N}_\alpha^p$ as well.The generating sequence of this power series space is of the form $(n^\theta)_{n\in\mathbb{N}}$ for some $0\lt\theta\lt1$. For example, the $\theta$s in the interval $(\smfr12,1)$ correspond in a bijective fashion to the Nevanlinna classes $\mathcal{N}_\alpha$, $\alpha\gt-1$, whereas the $\theta$s in the interval $(0,\smfr12)$ correspond bijectively to the Hardy–Orlicz spaces $LH^p$, $1\lt p\lt \infty$. By the work of Yanagihara, $\theta=\smfr12$ corresponds to $\mathcal{N}^+$.As in the work by Yanagihara, we derive our results from characterizations of coefficient multipliers from $\mathcal{N}_\alpha^p$ into various smaller classical spaces of analytic functions on $\Delta$.AMS 2000 Mathematics subject classification: Primary 46E10; 46A11; 47B38. Secondary 30D55; 46A45; 46E15\vskip-3pt