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Boman equals John. (English) Zbl 0861.43007

Laine, Ilpo (ed.) et al., 16th Rolf Nevanlinna colloquium. Proceedings of the international conference held in Joensuu, Finland, August 1–5, 1995. Berlin: de Gruyter. 91-99 (1996).
A homogeneous space \((S,d, \mu)\) has a pseudometric \(d\) and measure \(\mu\) that is doubling with respect to metric balls. A connected open set \(E\) in \(S\) is said to satisfy the Boman chain condition if there exist positive constants \(M\), \(\lambda > 1\), \(C_2 > C_1 > 1\), and a family \(F\) of disjoint metric balls \(B\) such that: (1) \(E = \cup_{B \in F} C_1B\); (2) \(\sum_{B \in F} \chi_{C_2B} (x) \leq M \chi_E(x)\) for all \(x \in S\); (3) there is a “central ball” \(B_*\) in \(F\) such that for each ball \(B\) in \(F\), there is a positive integer \(k\) and a chain of balls \(B_0 = B\), \(B_1, \dots, B_k = B_*\), and \(C_1B_j \cap C_1 B_{j+1}\) contains a metric ball \(D_j\) whose measure is comparable to those of both \(B_j\) and \(B_{j+1}\); (4) \(B \subset \lambda B_j\) for all \(j=0, \dots, k\). – The domain \(E\) is called a John domain with distinguished point \(x_*\) if there exists \(c>0\) such that for all \(x\) in \(E\), there is a curve \(\gamma\) in \(E\) parametrized by arclength beginning at \(x\) and ending at \(x_*\) such that \(d (\gamma (t), E^c) \geq ct\).
The authors prove that if \(E\) is a proper open subset of \(S\), then if \(E\) is a John domain, it is a Boman chain domain as well. The converse is true as well if it is assumed that \(E\) satisfies a strong geodesic condition.
For the entire collection see [Zbl 0852.00025].

MSC:

43A85 Harmonic analysis on homogeneous spaces
22E30 Analysis on real and complex Lie groups
31C45 Other generalizations (nonlinear potential theory, etc.)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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