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Duality for some large spaces of analytic functions


Jarchow, H; Montesinos, V; Wirths, K; Xiao, J (2001). Duality for some large spaces of analytic functions. Proceedings of the Edinburgh Mathematical Society (Series 2), 44(3):571-583.

Abstract


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 $\Delta$.

Abstract


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 $\Delta$.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:coefficient multipliers; weighted area Nevanlinna classes; weighted Bergman spaces; nuclear power series spaces
Language:English
Date:2001
Deposited On:29 Nov 2010 16:27
Last Modified:05 Apr 2016 13:25
Publisher:UNSPECIFIED
ISSN:0013-0915
Additional Information:Copyright: Cambridge University Press
Publisher DOI:https://doi.org/10.1017/S0013091599001042
Related URLs:http://www.ams.org/mathscinet-getitem?mr=1875769
http://www.zentralblatt-math.org/zbmath/search/?q=an%3A1001.46014

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