Abstract:
This article concerns algebras of $C^1$-functions in the disk $|z|<1$ such that $|\overline\partial f(z)|<w(1-|z|)$, where $w\uparrow$, and $\int_0\log\log w^{-1}(x)\,dx=+\infty$. For these functions a factorization theorem (on representation of each such function as the product of an analytic function and an antianalytic function, to within a function tending to zero as the boundary is approached) and a number of boundary uniqueness theorems are proved. One of these theorems is equivalent to a result generalizing the classical Levinson–Cartwright and Beurling theorems and consisting in the following. If $f(z)=\sum_{n<0}a_nz^n$, $|z|>1$, $|a_n|<e^{-p_n}$, $\sum_{n>0}p_n/n^2=\infty$, $F$ is analytic in the disk $|z|<1$, and $|F(z)|=o(w^{-1}(c(1-|z|)))$ as $|z|\to1$ for all $c<\infty$, where $w(x)=\exp(-\sup_n(p_n-nx))$, then $f=0$ and $F=0$ if $F$ has nontangential boundary values equal to the values of $f$ on some subset of the circle $|z|=1$ of positive Lebesgue measure. Here certain regularity conditions are imposed on $p$ and $w$. Uniqueness and factorization theorems for almost analytic functions are applied to the description of translation-invariant subspaces in the asymmetric algebras of sequences
$$
\mathfrak A=\{\{a_n\};\forall\,c\enskip\exists\,c_1:|a_n|<c_1e^{-cp_n},\ n<0,\ \exists\,c,\,\exists\,c_1:|a_n|<c_1e^{cp_n},\ n\geqslant0\}.
$$
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