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Some characterizations of Bloch functions on strongly pseudoconvex domains. (English) Zbl 0761.32005

The main result of the paper is the following theorem. Let \(D\) be a strongly pseudoconvex domain in \(\mathbb{C}^ n\) with defining function \(\rho\). Let \(F_ K^ D\), \(d_ K\) denote the Kobayashi-Royden metric and the Kobayashi distance for \(D\), respectively. Put \(B_ K(q,r):=\{z\in D\): \(d_ K(q,z)<r\}\). Let \(f\) be a holomorphic function on \(D\). Define \[ f_{B_ K(q,r)}:={1 \over {| B_ K(q,r)}} \int_{B_ K(q,r)} f(z)dw(z), \] where \(dw\) denotes the Euclidean volume element in \(\mathbb{C}^ n\). Then for any \(0<r,p<+\infty\) the following conditions are equivalent:
(i) \(f\) is a Bloch function, i.e. \[ \begin{aligned} &\sup\{| f'(z)X|/F_ K^ D(z,X):\;z\in D,\;X\in\mathbb{C}^ n\setminus \{0\}\}<+\infty,\\ &\sup_{q\in D}\left\{\left[{1 \over {| B_ K(q,r)|}} \int_{B_ K(q,r)} | f(z)-f(q)|^ p dw(z)\right]^{1/p}\right\}<+\infty,\tag{ii}\\ &\sup_{q\in D}\left\{\left[{1 \over {| B_ K(q,r)|}} \int_{B_ K(q,r)} | f(z)-f_{B_ K(q,r)}|^ p dw(z)\right]^{1/p}\right\}<+\infty,\tag{iii}\\ &\sup_{q\in D}\{\text{dist}(\bar f\mid_{B_ K(q,r)},H^ \infty(B_ K(q,r)))\}<+\infty,\tag{iv}\\ &\sup_{q\in D}\{\text{area}(f(B_ K(q,r)))\}<+\infty,\tag{v}\\ &\sup_{q\in D}\bigl\{\int_{B_ K(q,r)} |\nabla f(z)|^ p|\rho(z)|^{p-n- 1}dw(z)\bigr\}<+\infty.\tag{vi} \end{aligned} \] {}.

MSC:

32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
32T99 Pseudoconvex domains
32F45 Invariant metrics and pseudodistances in several complex variables
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