open access

https://hdl.handle.net/2115/43861

A uniqueness theorem and the myrberg phenomenon for a Zalcman domain

英語

2000

Springer

Journal d'Analyse Mathématique

82(1)

267-283

LetR=Δ0\∪nΔn be a Zalcman domain (or L-domain), where Δ0 : 0<|z| <1, Δn : |z-c n|≤r n,cn ↘0, Δn ⊂ Δ0 and Δn ∩ Δm= φ(n≠m). 0217 0115 V 3 For an unlimited two-sheeted covering R = φ-1 (R) with the branch points {φ-1(c n)}, set R = φ-1 (R). In the casec n=2−n , it was proved that if a uniqueness theorem is valid forH ∞ (R) atz=0, then the Myrberg phenomenon H∞(R) oφ = H∞ (R) occurs. One might suspect that the converse also holds. In this paper, contrary to this intuition, we show that the converse of this previous result is not true. In addition, we generalize the previous result for more general sequences {c n}. By this generalization we can even partly simplify the previous proof.

学術雑誌論文

Accepted Manuscript

The original publication is available at www.springerlink.com

AA00693150