Linear differential equations with coefficients in weighted Bergman and Hardy spaces

Complex linear differential equations of the form

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

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.