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Bounded mean oscillation of Bloch pull-backs. (English) Zbl 0727.32002

Given a holomorphic map F: \(B_ n\to D\), where \(B_ n\) denotes the open unit ball in \({\mathbb{C}}^ n\) and D denotes the open unit disk in \({\mathbb{C}}\), we say that F has the pull-back property if \(f\circ F\in BMOA(B_ n)\) whenever f belongs to the Bloch space of D. Ahern and Budin posed the problem of characterizing the maps F having the pull-back property. Previously only certain homogeneous polynomials were known to have the property. Here we show that F has the pull-back property whenever \(F\in Lip_ 1(B_ n)\). On the other hand, a result of Tomaszewski shows that there exist maps F failing to have the pull-back property even though \(F\in Lip_{\alpha}(B_ n)\) for some \(\alpha >0\).

MSC:

32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
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References:

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