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Zbl 0918.30011
Koskela, Pekka; MacManus, Paul
Quasiconformal mappings and Sobolev spaces.
(English)
[J] Stud. Math. 131, No.1, 1-17 (1998). ISSN 0039-3223; ISSN 1730-6337/e

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$(\int_B| u-a|^qd\mu)^{1/q}\le Cr(\int_{\lambda B}g^pd\mu)^{1/p}$. The function $g$ is usually the upper gradient of $u$, see [{\it 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 $1\le p\le Q$, 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 $1\le p'<Q$ for $1\le p<Q$. 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 $f:G\to\bbfR^n$, $G\subset\bbfR^n$ 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 {\it P. Hajłasz} [Potential Anal. 5, No. 4, 403-415 (1996; Zbl 0859.46022)] or due to {\it N. J. Korevaar} and {\it R. M. Schoen} [Commun. Anal. Geom. 1, No. 4, 561-659 (1993; Zbl 0862.58004)] can be used.
[O.Martio (Helsinki)]
MSC 2000:
*30C65 Quasiconformal mappings in R$\sp n$ and other generalizations
46E35 Sobolev spaces and generalizations

Keywords: quasiconformal mappings in metric spaces

Citations: Zbl 0870.54031; Zbl 0859.46022; Zbl 0862.58004

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