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Nevanlinna algebras

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The Nevanlinna algebras, $𝓝_{α}^{p}$, of this paper are the $L^{p}$ variants of classical weighted area Nevanlinna classes of analytic functions on 𝕌 = {z ∈ ℂ: |z| < 1}. They are F-algebras, neither locally bounded nor locally convex, with a rich duality structure.
For s = (α+2)/p, the algebra $F_{s}$ of analytic functions f: 𝕌 → ℂ such that $(1-|z|)^{s}|f(z)| → 0$ as |z| → 1 is the Fréchet envelope of $𝓝_{α}^{p}$. The corresponding algebra $𝓝_{s}^{∞}$ of analytic f: 𝕌 → ℂ such that $sup_{z∈𝕌} (1-|z|)^{s} |f(z)| < ∞$ is a complete metric space but fails to be a topological vector space. $F_{s}$ is also the largest linear topological subspace of $𝓝_{s}^{∞}$. $F_{s}$ is even a nuclear power series space. $𝓝_{α}^{p}$ and $𝓝_{β}^{q}$ generate the same Fréchet envelope iff (α+2)/p = (β+2)/q; they can replace each other for quasi-Banach space-valued continuous multilinear mappings.
Results for composition operators between $𝓝_{α}^{p}$'s can often be translated in a one-to-one fashion to corresponding ones on associated weighted Bergman spaces $𝓐_{α}^{p}$. This follows from the fact that the invertible elements in each $𝓝_{α}^{p}$ are precisely the exponentials of functions in $𝓐_{α}^{p}$. Moreover, each $𝓝_{α}^{p}$, (α+2)/p ≤ 1, admits dense ideals.
$𝓐_{α}^{p}$ embeds order boundedly into $𝓐_{β}^{q}$ iff $𝓐_{β}^{q}$ contains the Bloch type space $𝓐_{(α+2)/p}^{∞} $ iff (α+2)/p < (β+1)/q. In particular, $⋃_{p>0}𝓐_{α}^{p}$ and $⋂_{p>0}𝓐_{α}^{p}$ do not depend on the particular choice of α > -1. The first space is a nuclear space, a copy of the dual of the space of rapidly decreasing sequences; the second has properties much stronger than being a Schwartz space but fails to be nuclear.
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  • Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland
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  • Institut für Mathematik, Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland
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bwmeta1.element.bwnjournal-article-doi-10_4064-sm147-3-4
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