Illinois Journal of Mathematics

On analytic and meromorphic functions and spaces of $Q\sb K$-type

Matts Essén and Hasi Wulan

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Starting from a nondecreasing function $K:[0,\infty)\to [0,\infty)$, we introduce a M\"obius-invariant Banach space $Q_K$ of functions analytic in the unit disk in the plane. We develop a general theory of these spaces, which yields new results and also, for special choices of $K$, gives most basic properties of $Q_p$-spaces. We have found a general criterion on the kernels $K_1$ and $K_2$, $K_1\leq K_2$, such that $Q_{K_2}\subsetneqq Q_{K_1}$, as well as necessary and sufficient conditions on $K$ so that $Q_K=\mathcal{B}$ or $Q_K =\mathcal{D}$, where the Bloch space $\mathcal{B}$ and the Dirichlet space $\mathcal{D}$ are the largest, respectively smallest, spaces of $Q_K$-type. We also consider the meromorphic counterpart $Q_K^\#$ of $Q_K$ and discuss the differences between $Q_K$-spaces and $Q_K^\#$-classes.

Article information

Illinois J. Math., Volume 46, Number 4 (2002), 1233-1258.

First available in Project Euclid: 13 November 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30D45: Bloch functions, normal functions, normal families
Secondary: 30D50 46E15: Banach spaces of continuous, differentiable or analytic functions


Essén, Matts; Wulan, Hasi. On analytic and meromorphic functions and spaces of $Q\sb K$-type. Illinois J. Math. 46 (2002), no. 4, 1233--1258. doi:10.1215/ijm/1258138477.

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