## 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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