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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
    • Article
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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