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CHARACTERISATIONS OF HARDY GROWTH SPACES WITH DOUBLING WEIGHTS

Part of: Harmonic analysis in one variable Spaces and algebras of analytic functions

Published online by Cambridge University Press:  12 May 2014

EVGUENI DOUBTSOV
EVGUENI DOUBTSOV*
Affiliation:
St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia email dubtsov@pdmi.ras.ru
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Abstract

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Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H(\mathbb{D})$ denote the space of holomorphic functions on the unit disc $\mathbb{D}$. Given $p>0$ and a weight $\omega $, the Hardy growth space $H(p, \omega )$ consists of those $f\in H(\mathbb{D})$ for which the integral means $M_p(f,r)$ are estimated by $C\omega (r)$, $0<r<1$. Assuming that $p>1$ and $\omega $ satisfies a doubling condition, we characterise $H(p, \omega )$ in terms of associated Fourier blocks. As an application, extending a result by Bennett et al. [‘Coefficients of Bloch and Lipschitz functions’, Illinois J. Math. 25 (1981), 520–531], we compute the solid hull of $H(p, \omega )$ for $p\ge 2$.

Keywords

Hardy growth spacedoubling weight

MSC classification

Secondary: 30H10: Hardy spaces 42A55: Lacunary series of trigonometric and other functions; Riesz products
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 90 , Issue 2 , October 2014 , pp. 275 - 282
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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