| Title | A conjecture of L. Carleson and applications |
| Authors | Alexandru Aleman |
| Alternative Location | http://dx.doi.org/10.1307/m... |
| Publication | Michigan mathematical journal |
| Year | 1992 |
| Volume | 39 |
| Issue | 3 |
| Pages | 537 - 549 |
| Document type | Article |
| Status | Published |
| Quality controlled | Yes |
| Language | eng |
| Publisher | University of Michigan, Department of Mathematics |
| Abstract English | 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> 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 $T_\alpha$. |
| ISBN/ISSN/Other | ISSN: 00262285 |
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