Linear differential equations with coefficients in weighted Bergman and Hardy spaces

Abstract:

Complex linear differential equations of the form \begin{equation}\tag {\dagger } f^{(k)}+a_{k-1}(z)f^{(k-1)}+\cdots +a_1(z)f’+a_0(z)f=0 \end{equation} with coefficients in weighted Bergman or Hardy spaces are studied. It is shown, for example, that if the coefficient $a_j(z)$ of $(\dagger )$ belongs to the weighted Bergman space $A^\frac {1}{k-j}_\alpha$, where $\alpha \ge 0$, 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 $\alpha \ge 0$, 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 $(\dagger )$ is also briefly discussed.