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.
