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harmonic besov space

Proceedings of the Edinburgh Mathematical Society (1)

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A CHARACTERIZATION OF THE HARMONIC BLOCH SPACE AND THE HARMONIC BESOV SPACES BY AN OSCILLATION
Rikio Yoneda
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 45 / Issue 1 / February 2002

Published online by Cambridge University Press:

05 February 2002, pp. 229-239
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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