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Keywords:
Bloch norm; Möbius transformation
Bloch norm; Möbius transformation
Summary:
For a $C^1$-function $f$ on the unit ball $\mathbb B \subset \mathbb C ^n$ we define the Bloch norm by $\|f\|_\mathfrak B=\sup \|\tilde df\|,$ where $\tilde df$ is the invariant derivative of $f,$ and then show that $$ \|f\|_\mathfrak B= \sup _{z,w\in {\mathbb B} \atop z\neq w} (1-|z|^2)^{1/2}(1-|w|^2)^{1/2}\frac {|f(z)-f(w)|}{|w-P_wz-s_wQ_wz|}.$$
References:
[1] Holland, F., Walsh, D.: Criteria for membership of Bloch space and its subspace, BMOA. Math. Ann. 273 (1986), 317-335. DOI 10.1007/BF01451410 | MR 0817885 | Zbl 0561.30025
[2] Nowak, M.: Bloch space and Möbius invariant Besov spaces on the unit ball of {${\mathbb C}^n$}. Complex Variables Theory Appl. 44 (2001), 1-12. DOI 10.1080/17476930108815339 | MR 1826712
[3] Pavlovi'c, M.: On the Holland-Walsh characterization of Bloch functions. Proc. Edinb. Math. Soc. 51 (2008), 439-441. DOI 10.1017/S0013091506001076 | MR 2465917
[4] Rudin, W.: Function Theory in the Unit Ball of {$C^n$}. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York (1980). MR 0601594
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