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Linear differential equations with coefficients in weighted Bergman and Hardy spaces

Authors:	Janne Heittokangas Risto Korhonen Jouni Rä ttyä

Institution:	Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, Illinois 61801 ; Department of Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland. ; Department of Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland

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 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 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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