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A conjecture of L. Carleson and applications

Aleman, Alexandru LU (1992) In Michigan Mathematical Journal 39(3). p.537-549
Abstract
Let $T_\alpha$ be the class of functions meromorphic in the unit disk ${\bf D}$ such that $$\int_D{|f'(z)|^2\over (1+|f(z)|^2)^2}(1-|z|)^{1-\alpha}dx\,dy<\infty,\quad 0\leq\alpha<1.$$



It is known that $T_\alpha\subset N$, where $N$ denotes the Nevanlinna class of functions meromorphic in $\bold D$ and of bounded characteristic. Because every $f\in N$ is a quotient of two bounded functions, analytic in ${\bf D}$, it is natural to ask whether any $f\in T_\alpha$ may be represented as a quotient of two bounded analytic functions of the same class. This problem was posed for the first time by Carleson in his thesis [``On a class of meromorphic functions and its associated exceptional sets'', Univ. Uppsala, Uppsala,... (More)
Let $T_\alpha$ be the class of functions meromorphic in the unit disk ${\bf D}$ such that $$\int_D{|f'(z)|^2\over (1+|f(z)|^2)^2}(1-|z|)^{1-\alpha}dx\,dy<\infty,\quad 0\leq\alpha<1.$$



It is known that $T_\alpha\subset N$, where $N$ denotes the Nevanlinna class of functions meromorphic in $\bold D$ and of bounded characteristic. Because every $f\in N$ is a quotient of two bounded functions, analytic in ${\bf D}$, it is natural to ask whether any $f\in T_\alpha$ may be represented as a quotient of two bounded analytic functions of the same class. This problem was posed for the first time by Carleson in his thesis [``On a class of meromorphic functions and its associated exceptional sets'', Univ. Uppsala, Uppsala, 1950; MR0033354 (11,427c)]. Carleson proved that any $f\in T_\alpha$ is a quotient of bounded functions $u,v\in T_\beta$, analytic in ${\bf D}$, for all $\beta<\alpha$. The considerable success of the author in the paper under review is in proving the following result. Theorem: For $0<\alpha\leq 1$, every function in $T_\alpha$ is a quotient of two bounded analytic functions belonging to $T_\alpha$. (Less)
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author
publishing date
type
Contribution to journal
publication status
published
subject
in
Michigan Mathematical Journal
volume
39
issue
3
pages
537 - 549
publisher
University of Michigan, Department of Mathematics
ISSN
0026-2285
DOI
10.1307/mmj/1029004605
language
English
LU publication?
no
id
9135b4f6-726b-46f8-9356-7048e5dc5bf0 (old id 1467365)
date added to LUP
2009-09-16 13:41:43
date last changed
2016-06-29 09:04:21
@article{9135b4f6-726b-46f8-9356-7048e5dc5bf0,
  abstract     = {Let $T_\alpha$ be the class of functions meromorphic in the unit disk ${\bf D}$ such that $$\int_D{|f'(z)|^2\over (1+|f(z)|^2)^2}(1-|z|)^{1-\alpha}dx\,dy&lt;\infty,\quad 0\leq\alpha&lt;1.$$ <br/><br>
<br/><br>
It is known that $T_\alpha\subset N$, where $N$ denotes the Nevanlinna class of functions meromorphic in $\bold D$ and of bounded characteristic. Because every $f\in N$ is a quotient of two bounded functions, analytic in ${\bf D}$, it is natural to ask whether any $f\in T_\alpha$ may be represented as a quotient of two bounded analytic functions of the same class. This problem was posed for the first time by Carleson in his thesis [``On a class of meromorphic functions and its associated exceptional sets'', Univ. Uppsala, Uppsala, 1950; MR0033354 (11,427c)]. Carleson proved that any $f\in T_\alpha$ is a quotient of bounded functions $u,v\in T_\beta$, analytic in ${\bf D}$, for all $\beta&lt;\alpha$. The considerable success of the author in the paper under review is in proving the following result. Theorem: For $0&lt;\alpha\leq 1$, every function in $T_\alpha$ is a quotient of two bounded analytic functions belonging to $T_\alpha$.},
  author       = {Aleman, Alexandru},
  issn         = {0026-2285},
  language     = {eng},
  number       = {3},
  pages        = {537--549},
  publisher    = {University of Michigan, Department of Mathematics},
  series       = {Michigan Mathematical Journal},
  title        = {A conjecture of L. Carleson and applications},
  url          = {http://dx.doi.org/10.1307/mmj/1029004605},
  volume       = {39},
  year         = {1992},
}