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Quasiconformal mappings and Sobolev spaces. (English) Zbl 0918.30011
The paper is devoted to the study how Poincaré inequalities change under quasisymmetric mappings between metric spaces. Suppose that a metric space $$X$$ is locally compact and equipped with a $$Q$$-regular Borel measure $$\mu$$. A pair $$(u,g)$$ of measurable functions in $$X$$ satisfies a $$(q,p)$$-Poincaré inequality if there are constants $$C$$ and $$\lambda$$ such that for each ball $$B$$ of radius $$r$$ there is a real number a such that$$(\int_B| u-a|^qd\mu)^{1/q}\leq Cr(\int_{\lambda B}g^pd\mu)^{1/p}$$. The function $$g$$ is usually the upper gradient of $$u$$, see [S. Semmens: Sel. Math. New Ser. 2, 155-295 (1996; Zbl 0870.54031)]. The main result says that if $$X$$ supports an $$(1,p)$$-Poincaré inequality for some $$1\leq p\leq Q$$, then given a quasisymmetric map $$f:X\to Y$$ the space $$Y$$ supports an $$(1,p')$$-Poincaré inequality where $$p'=p$$ if $$p=Q$$ and $$1\leq p'<Q$$ for $$1\leq p<Q$$. In the space $$Y$$ the pullback measure of $$\mu$$ under $$f$$ is used. The authors also show that the result does not hold for $$p>Q$$ in general. Extending the classical result that the Dirichlet space $$L^{1,n}(G)$$ is preserved under a quasiconformal mapping $$f:G\to\mathbb{R}^n$$, $$G\subset\mathbb{R}^n$$ a domain, they prove that the corresponding Dirichlet spaces in the metric setup for $$n = Q$$ are preserved as well. Here the definitions of the first order Sobolev spaces on a metric space due to P. Hajłasz [Potential Anal. 5, No. 4, 403-415 (1996; Zbl 0859.46022)] or due to N. J. Korevaar and R. M. Schoen [Commun. Anal. Geom. 1, No. 4, 561-659 (1993; Zbl 0862.58004)] can be used.

##### MSC:
 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
##### Keywords:
quasiconformal mappings in metric spaces
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