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Quasiconformal maps in metric spaces with controlled geometry. (English) Zbl 0915.30018
For a long time it has been an open problem in the quasiconformal mapping theory to prove, by simple geometric means, that the local quasiconformality condition implies the global condition. This means the following: If a homeomorphism $$f:D\to\mathbb{R}^n$$, $$D\subset\mathbb{R}^n$$ a domain, satisfies $$H_f(x)\leq H$$ for all $$x\in D$$, then it satisfies a global or semiglobal quasisymmetry condition $$H_f(x,r)\leq H'$$. Here $$H_f(x,r)= L(x,f,r)/l(x,f,r)$$, $$0<r<d(x,\partial D)$$, and $$H_f(x)=\limsup_{r\to 0}H_f(x,r)$$. Now these conditions make perfect sense for homeomorphisms $$f:X\to Y$$ between metric spaces $$X$$ and $$Y$$. The authors show that the pointwise condition on $$H_f(x)$$ is quantitatively equivalent to the global or semiglobal quasisymmetry condition in a large class of metric spaces, including $$\mathbb{R}^n$$ and Carnot-Carathéodory spaces. A natural assumption is that $$X$$ and $$Y$$ are equipped with measures which fit together with the metric structure – $$Q$$-regular spaces: $$C^{-1}\mathbb{R}^Q\leq \mu(B(x,\mathbb{R})) \leq C\mathbb{R}^Q$$, $$Q>1$$. In [J. Heinonen and P. Koskela, Invent Math. 120, No. 1, 61-79 (1995; Zbl 0832.30013)] a discrete modulus was employed but here it is replaced by a continuous approach; for this the conecpt of the weak gradient S. Semmes, Publ. Math. 40, 4ll–430] is used. A central idea is to study Loewner spaces, i.e., spaces which have a lower bound for the modulus (or capacity) of a curve family connecting two continua in terms of the diameters and the distance of the continua. The modulus is the $$p$$-modulus with $$p$$ = Hausdorff dimension of $$X$$. Now the conclusion from the pointwise condition to the global condition follows provided that $$X$$ is a Loewner spaee and $$Y$$ is linearly locally connected. The Loewner condition is closely related to the Poincaré inequality. Absolute continuity of quasiconformal mappings is also studied in the metric setup.

##### MSC:
 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations
##### Keywords:
quasiconformal mapping; Carnot-Carathéodory space
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