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Exact location of \(\alpha\)-Bloch spaces in \(L^p_a\) and \(H^p\) of a complex unit ball. (English) Zbl 0978.32002

We say that \(f\in\mathcal B^\alpha\), \(\alpha\)-Bloch, if \[ \|f\|_{\mathcal B^\alpha(B)} = \sup_{z\in B}|\nabla f(z)|(1 - |z|^2)^\alpha <\infty, \quad 0 < \alpha < \infty, \] where \(B\) denotes the unit ball in \(\mathbb C^n\).
The authors investigate the relationships between \(\alpha\)-Bloch and some classes of holomorphic functions, such as the exact location of \(\alpha\)-Bloch spaces in the Bergman spaces \(L^p_a\) and Hardy spaces \(H^p\). It is proved that (i) \(f\in\mathcal B^\alpha\Leftrightarrow \sup_{z\in B}|\mathcal R f(z)|(1 - |z|^2)^\alpha <\infty\), \(\mathcal B^\alpha = A(B)\cap \text{Lip}(1 - \alpha)\) for \(0 < \alpha < 1\), where \(A(B)\) is the ball algebra. (ii) \(\mathcal B^{\alpha(<1+1/p)}\subset L_a^p\subset\mathcal B^{1+(n+1)/p}\), \(\mathcal B^{\alpha(<1)} \subset H^p \subset\mathcal B^{1+n/p}\) for \(n > 1\) and \(0 < p < \infty\). Further, \(\mathcal B^1 \subset \bigcap _{0<p<\infty}L_a^p\subset\mathcal B^{\alpha(>1)}\). \(\mathcal B^{\alpha(<1)} \subset \bigcap _{0<p<\infty}H^p\subset\mathcal B^{\alpha(>1)}\). All of the containments are strict and best possible. For the inclusion chain \(\mathcal B^{\alpha(<1+1/p)}\subset L_a^p\subset \mathcal B^{1+(n+1)/p}\), the strictness at the left side and the possibility at the right side show that, for each \(p\), at least one \(f(z)\) exists, \(f\in L_a^p\), whose growth rate of gradient, or radial derivative, will be larger than, or equal to, \((1 - |z|^2)^{-(1+1/p)}\), and go so far as to \((1 - |z|^2)^{-(1+(n+1)/p)}\). There is a similar conclusion for \(H^p\) in the other inclusion chain. Especially in the proof of the strictness and best possibility in (ii), constructive methods are used.

MSC:

32A18 Bloch functions, normal functions of several complex variables
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
46E99 Linear function spaces and their duals
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References:

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