A conjecture of L. Carleson and applications
(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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http://lup.lub.lu.se/record/1467365
- author
- Aleman, Alexandru ^{LU}
- publishing date
- 1992
- 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 $$\int_D{|f'(z)|^2\over (1+|f(z)|^2)^2}(1-|z|)^{1-\alpha}dx\,dy<\infty,\quad 0\leq\alpha<1.$$ <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 $\beta<\alpha$. 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 $T_\alpha$.}, 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}, }