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Abstract

The Nevanlinna algebras, ${\cal N}_\alpha ^p$, of this paper are the $L^p$ variants of classical weighted area Nevanlinna classes of analytic functions on ${\mathbb U}=\{ z\in {\mathbb C} :|z|<1\} $. They are $F$-algebras, neither locally bounded nor locally convex, with a rich duality structure.

For $s={(\alpha +2)/p}$, the algebra $F_s$ of analytic functions $f:{\mathbb U}\to {\mathbb C}$ such that $(1-|z|)^s|f(z)|\to 0$ as $|z|\to 1$ is the Fréchet envelope of ${\cal N}_\alpha ^p$. The corresponding algebra ${\cal N}_s^\infty $ of analytic $f:{\mathbb U}\to {\mathbb C}$ such that $\mathop {\rm sup}_{z\in {\mathbb U}}(1-|z|)^s|f(z)|<\infty $ is a complete metric space but fails to be a topological vector space. $F_s$ is also the largest linear topological subspace of ${\cal N}_s^\infty $. $F_s$ is even a nuclear power series space. ${\cal N}_\alpha ^p$ and ${\cal N}_\beta ^q$ generate the same Fréchet envelope iff ${(\alpha +2)/p}={(\beta +2)/q}$; they can replace each other for quasi-Banach space-valued continuous multilinear mappings.

Results for composition operators between ${\cal N}_\alpha ^p$'s can often be translated in a one-to-one fashion to corresponding ones on associated weighted Bergman spaces ${\cal A}_\alpha ^p$. This follows from the fact that the invertible elements in each ${\cal N}_\alpha ^p$ are precisely the exponentials of functions in ${\cal A}_\alpha ^p$. Moreover, each ${\cal N}_\alpha ^p$, ${(\alpha +2)/p}\le 1$, admits dense ideals.

${\cal A}_\alpha ^p$ embeds order boundedly into ${\cal A}_\beta ^q$ iff ${\cal A}_\beta ^q$ contains the Bloch type space ${\cal A}_{(\alpha +2)/p}^\infty $ iff ${(\alpha +2)/p}<(\beta +1)/q$. In particular, $\bigcup _{p>0}{\cal A}_\alpha ^p$ and $\bigcap _{p>0}{\cal A}_\alpha ^p$ do not depend on the particular choice of $\alpha >-1$. 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.