Linear differential equations with coefficients in weighted Bergman and Hardy spaces

Abstract

Complex linear differential equations of the form (\dagger) f ( k ) + a k − 1 ( z ) f ( k − 1 ) + ⋯ + a 1 ( z ) f ′ + a 0 ( z ) f = 0 \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 ) a_j(z) of ( † ) (\dagger ) belongs to the weighted Bergman space A α 1 k − j A^\frac {1}{k-j}_\alpha , where α ≥ 0 \alpha \ge 0 , for all j = 0 , … , k − 1 j=0,\ldots ,k-1 , then all solutions are of order of growth at most α \alpha , measured according to the Nevanlinna characteristic. In the case when α = 0 \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 α ≥ 0 \alpha \ge 0 , then the coefficient a j ( z ) a_j(z) is shown to belong to A α p j A^{p_j}_\alpha for all p j ∈ ( 0 , 1 k − j ) p_j\in (0,\frac {1}{k-j}) and j = 0 , … , k − 1 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.