Linear differential equations with solutions in the Dirichlet type subspace of the Hardy space

Abstract

Sufficient conditions for the analytic coefficients of the linear differential equation

f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_{1}(z)f'+A_{0}(z)f = 0

are found such that all solutions belong to a given $H^{\infty}_{q}$-space, or to the Dirichlet type subspace $\mathcal{D}^{p}$ of the classical Hardy space $H^{p}$, where $0 < p \le 2$. For $0 < q < \infty$, the space $H^{\infty}_{q}$ consists of those functions $f$, analytic in the unit disc $D$, for which $|f(z)|(1-|z|^{2})^{q}$ is uniformly bounded in $D$, and $f \in \D^{p}$ if the integral $\int_{D} |f'(z)|^{p}(1-|z|^{2})^{p-1} \, d\sigma_{z}$ converges.