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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
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