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A CHARACTERIZATION OF THE HARMONIC BLOCH SPACE AND THE HARMONIC BESOV SPACES BY AN OSCILLATION

Published online by Cambridge University Press:  05 February 2002

Rikio Yoneda
Rikio Yoneda
Affiliation:
Tokyo Metropolitan College of Technology, Tokyo 140-0011, Japan ([email protected])
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Abstract

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We characterize the Bloch space and the Besov spaces of harmonic functions on the open unit disc $D$ by using the following oscillation:

$$ \sup_\{\beta(z,w)\ltr\}(1-|z|^2)^{\alpha}(1-|w|^2)^{\beta}\biggl|\frac{\hat{D}^{(n-1)}h(z)-\hat{D}^{(n-1)}h(w)}{z-w}\biggr|, $$

where $\alpha+\beta=n$, $\alpha,\beta\in\mathbb{R}$ and $\displaystyle{\hat{D}^{(n)}=(\partial^{n}/\partial^{n}z+\partial^{n}/\partial^{n}\bar{z})}$.

AMS 2000 Mathematics subject classification: Primary 46E15

Keywords

harmonic Bloch spaceharmonic Besov spaceoscillation
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 45 , Issue 1 , February 2002 , pp. 229 - 239
Copyright
Copyright © Edinburgh Mathematical Society 2002