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2007 Linear differential equations with solutions in the Dirichlet type subspace of the Hardy space
J. Heittokangas, R. Korhonen, J. Rättyä
Nagoya Math. J. 187: 91-113 (2007).

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.

Citation

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J. Heittokangas. R. Korhonen. J. Rättyä. "Linear differential equations with solutions in the Dirichlet type subspace of the Hardy space." Nagoya Math. J. 187 91 - 113, 2007.

Information

Published: 2007
First available in Project Euclid: 4 September 2007

zbMATH: 1161.34060
MathSciNet: MR2354557

Subjects:
Primary: 34M10
Secondary: 30D50 , 30D55

Rights: Copyright © 2007 Editorial Board, Nagoya Mathematical Journal

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