On Lipschitz and α-Bloch spaces on the unit polydisc
Authors: Stević Stevo 
Published: September 2008

In this paper we show that if α ∈ (0, 1), the holomorphic Lipschitz space
Lα
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$$\mathbb{D}^n$$
\end{document}) on the unit polydisc \documentclass{aastex}
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$$\mathbb{D}^n$$
\end{document} is equivalent to the 1 − α-Bloch space \documentclass{aastex}
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$$\mathcal{B}^{1 - \alpha } (\mathbb{D}^n )$$
\end{document}, as well as to the space consisting of all holomorphic functions
f
on \documentclass{aastex}
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$$\mathbb{D}^n$$
\end{document}, continuous on the closure of \documentclass{aastex}
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$$\mathbb{D}^n$$
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$$\mathop {sup}\limits_{\theta ,\phi \in [0,2\pi ]^n ,\theta \ne \phi } \frac{{|f(e^{i\theta } ) - f(e^{i\phi } )|}}{{|\theta - \phi |^\alpha }} < \infty ,$$
\end{document} where
eiϑ
is an abbreviation for (\documentclass{aastex}
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$$e^{i\theta _1 } ,...,e^{i\theta _n }$$
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$$\left\| \cdot \right\|_{L_\alpha } ,\left\| \cdot \right\|_{\mathcal{B}^{1 - \alpha } }$$
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$$\left\| \cdot \right\|_{\Lambda _\alpha }$$
\end{document}, on these spaces are comparable, that is, there are positive constants
C1
,
C2
and
C3
depending only on
α
and
n
such that \documentclass{aastex}
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$$\left\| f \right\|_{L_\alpha } \leqq C_1 \left\| f \right\|_{\mathcal{B}^{1 - \alpha } } \leqq C_2 \left\| f \right\|_{\Lambda _\alpha } \leqq C_3 \left\| f \right\|_{L_\alpha } ,$$
\end{document} for every
f
holomorphic on \documentclass{aastex}
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$$\mathbb{D}^n$$
\end{document}.