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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Linear differential equations with coefficients in weighted Bergman and Hardy spaces


Authors: Janne Heittokangas, Risto Korhonen and Jouni Rättyä
Journal: Trans. Amer. Math. Soc. 360 (2008), 1035-1055
MSC (2000): Primary 34M10; Secondary 30D50, 30D55
Published electronically: August 20, 2007
MathSciNet review: 2346482
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Abstract: Complex linear differential equations of the form

$\displaystyle (\dag )\qquad\qquad\qquad f^{(k)}+a_{k-1}(z)f^{(k-1)}+\cdots +a_1(z)f'+a_0(z)f=0 \qquad\qquad $

with coefficients in weighted Bergman or Hardy spaces are studied. It is shown, for example, that if the coefficient $ a_j(z)$ of $ (\dag )$ belongs to the weighted Bergman space $ A^\frac{1}{k-j}_\alpha$, where $ \alpha\ge0$, for all $ j=0,\ldots,k-1$, then all solutions are of order of growth at most $ \alpha$, measured according to the Nevanlinna characteristic. In the case when $ \alpha=0$ all solutions are shown to be not only of order of growth zero, but of bounded characteristic. Conversely, if all solutions are of order of growth at most $ \alpha\ge0$, then the coefficient $ a_j(z)$ is shown to belong to $ A^{p_j}_\alpha$ for all $ p_j\in(0,\frac{1}{k-j})$ and $ j=0,\ldots,k-1$.

Analogous results, when the coefficients belong to certain weighted Hardy spaces, are obtained. The non-homogeneous equation associated to $ (\dag )$ is also briefly discussed.


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Additional Information

Janne Heittokangas
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801
Address at time of publication: Department of Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland
Email: janne.heittokangas@joensuu.fi

Risto Korhonen
Affiliation: Department of Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland.
Email: risto.korhonen@joensuu.fi

Jouni Rättyä
Affiliation: Department of Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland
Email: jouni.rattya@joensuu.fi

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04335-8
PII: S 0002-9947(07)04335-8
Received by editor(s): August 15, 2005
Received by editor(s) in revised form: March 15, 2006
Published electronically: August 20, 2007
Additional Notes: This research was supported in part by the Academy of Finland 204819 and 210245, MEC Spain MTM2004-21420-E, and the Väisälä Fund of the Finnish Academy of Science and Letters.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.