DUALITY FOR SOME LARGE SPACES OF ANALYTIC FUNCTIONS

• Published in: Proceedings of the Edinburgh Mathematical Society Vol. 44; no. 3; pp. 571 - 583
• Main Authors: Jarchow, H., Montesinos, V., Wirths, K. J., Xiao, J.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.10.2001
• Subjects: coefficient multipliers; nuclear power series spaces; weighted area Nevanlinna classes; weighted Bergman spaces; coefficient multipliers; nuclear power series spaces; weighted Bergman spaces; weighted area Nevanlinna classes
• ISSN: 0013-0915; 1464-3839
• DOI: 10.1017/S0013091599001042

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