## Nagoya Mathematical Journal

### 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.

#### Article information

Source
Nagoya Math. J., Volume 187 (2007), 91-113.

Dates
First available in Project Euclid: 4 September 2007

https://projecteuclid.org/euclid.nmj/1188913896

Mathematical Reviews number (MathSciNet)
MR2354557

Zentralblatt MATH identifier
1161.34060

Subjects
Primary: 34M10: Oscillation, growth of solutions
Secondary: 30D50 30D55

#### Citation

Heittokangas, J.; Korhonen, R.; Rättyä, J. Linear differential equations with solutions in the Dirichlet type subspace of the Hardy space. Nagoya Math. J. 187 (2007), 91--113. https://projecteuclid.org/euclid.nmj/1188913896

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